\subsection{Pairs that are holomorphic past the circle are not extremal}\label{sec:THMS}

Lemma~\ref{lem:ES} needs $W$ univalent past the circle. Here $W$ need only be holomorphic past the circle: it may touch itself on the
circle or have critical points there. Let $\mathbb T=\partial\mathbb D$. For $W\in\mathcal S$ holomorphic on an open set
$U\supset\overline{\mathbb D}$ put
\[
  \mathrm{Crit}=\{\zeta\in\mathbb T:W'(\zeta)=0\},\qquad
  \mathrm{Cont}=\{\zeta\in\mathbb T:W(\zeta)=W(\zeta')\text{ for some }\zeta'\in\mathbb T\setminus\{\zeta\}\}.
\]
We call $\zeta'$ a \emph{partner} of $\zeta$, and $\{\zeta,\zeta'\}$ a \emph{contact pair}\label{term:crit}.
$W$ is univalent on a neighbourhood of the closed disk if and only if $\mathrm{Crit}\cup\mathrm{Cont}$ is empty (by compactness and
part~(a) of the next lemma); that is the case of Lemma~\ref{lem:ES}.

\begin{lemma}\label{lem:D}
Let $W\in\mathcal S$ be holomorphic on an open set $U\supset\overline{\mathbb D}$.
\begin{enumerate}
\item[(a)] $W(\mathbb T)\cap W(\mathbb D)=\emptyset$.
\item[(b)] Every $\zeta\in\mathrm{Crit}$ is a simple zero of $W'$, and $W(\zeta')\ne W(\zeta)$ for $\zeta'\in\mathbb T\setminus\{\zeta\}$.
\item[(c)] Every $\zeta\in\mathrm{Cont}$ has exactly one partner $\zeta'$, and $W'(\zeta)\ne0\ne W'(\zeta')$.
\item[(d)] $\mathrm{Crit}$ and $\mathrm{Cont}$ are finite.
\item[(e)] Each $c\in\mathrm{Crit}$ has a \emph{partner map} $\beta$: a holomorphic involution of a neighbourhood of $c$ with
$\beta(c)=c$ such that, for $w\ne w'$ near $c$, $W(w)=W(w')$ if and only if $w'=\beta(w)$. Each contact pair
$\{\zeta_1,\zeta_2\}$ has a partner map $\tau$: a conformal map of a neighbourhood of $\zeta_1$ onto one of $\zeta_2$ with
$\tau(\zeta_1)=\zeta_2$ and $W\circ\tau=W$. The \emph{gap functions} $\operatorname{gap}(s)=\log|\beta(ce^{is})|$ and
$\operatorname{gap}(s)=\log|\tau(\zeta_1e^{is})|$ are real-analytic near $0$, with $\operatorname{gap}(0)=0$ and
$\operatorname{gap}(s)\ge c_{\mathrm{gap}}s^{2j}$ near $0$ for some $c_{\mathrm{gap}}>0$ and an integer $j\ge1$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) If $W(\zeta)=W(w_0)$ with $\zeta\in\mathbb T$, $w_0\in\mathbb D$, take a disk $\Delta_0$ about $w_0$ with closure in $\mathbb D$, not
containing $\zeta$ in its closure. $W(\Delta_0)$ is a neighbourhood of $W(w_0)$, so points $w\in\mathbb D\setminus\overline{\Delta_0}$ near
$\zeta$ have $W(w)=W(\tilde w)$ with $\tilde w\in\Delta_0$, $\tilde w\ne w$, against injectivity on $\mathbb D$.

(b), (c) Near $\zeta\in\mathbb T$ with $W'(\zeta)\neq0$, $W(\mathbb D\cap\Delta)$ contains a sector at $W(\zeta)$ of opening
$\pi-\varepsilon$ for small disks $\Delta$ about $\zeta$. If $W-W(\zeta)$ has a zero of order $q+1$ at $\zeta$, write $W-W(\zeta)=\hat x^{q+1}$ with $\hat x$ conformal at $\zeta$,
$\hat x(\zeta)=0$; $\hat x(\mathbb D\cap\Delta)$ contains a sector at $0$ of opening $\pi-\varepsilon$, and $u\mapsto u^{q+1}$ is not
injective on a sector of opening larger than $2\pi/(q+1)$; so $q\le1$. For $q=1$, $W(\mathbb D\cap\Delta)$ contains a sector of opening
$2\pi-2\varepsilon$. The sets $W(\mathbb D\cap\Delta)$ for disjoint small disks $\Delta$ about the preimages of one value are disjoint.
Two sectors of opening $\pi-\varepsilon$ fit at one vertex, three do not, and none fits beside one of opening $2\pi-2\varepsilon$.

(d), (e) \emph{Local maps.} At $c\in\mathrm{Crit}$ write $W=W(c)+\hat x^2$ with $\hat x$ conformal near $c$, $\hat x(c)=0$; then
$\beta=\hat x^{-1}(-\hat x)$ is a holomorphic involution near $c$ with $\beta(c)=c$, $\beta'(c)=-1$, and for $w\ne w'$ near $c$,
$W(w)=W(w')$ if and only if $w'=\beta(w)$. At a pair $\{\zeta_1,\zeta_2\}$ of $\mathrm{Cont}$ let
$\tau=(W|_{\Delta_2})^{-1}\circ W$ near $\zeta_1$, where $W$ is conformal on disjoint disks $\Delta_1,\Delta_2$ about $\zeta_1,\zeta_2$;
$\tau(\zeta_1)=\zeta_2$. Since $W$ is injective on $\mathbb D$, $|\beta|\ge1$ (resp. $|\tau|\ge1$) on $\overline{\mathbb D}$ near $c$
(resp. $\zeta_1$), so $\operatorname{gap}(s)=\log|\beta(ce^{is})|$ (resp. $\log|\tau(\zeta_1e^{is})|$) is real-analytic near $0$, $\operatorname{gap}\ge0$, $\operatorname{gap}(0)=0$.

\emph{Reduction to arcs.} $\mathrm{Crit}$ is finite, as $W'\not\equiv0$. If some gap function vanishes identically near $0$, then $\beta$ or $\tau$ maps an arc
of $\mathbb T$ into $\mathbb T$, and $\mathrm{Cont}$ contains an arc. If $\mathrm{Cont}$ is infinite, contact pairs
$(\zeta_n,\zeta_n')$ accumulate at some $(\zeta^*,\zeta'^*)$ with $W(\zeta^*)=W(\zeta'^*)$. If $\zeta^*\ne\zeta'^*$ this is a contact pair
and $\zeta_n'=\tau(\zeta_n)$ eventually; if $\zeta^*=\zeta'^*$, then $\zeta^*\in\mathrm{Crit}$ (else $W$ is injective near $\zeta^*$) and
$\zeta_n'=\beta(\zeta_n)$. Either way the gap function has zeros $s_n\to0$, $s_n\ne0$, so it vanishes identically. It remains to show that $\mathrm{Cont}$ contains
no arc; then (d) holds, and (e) as well: a real-analytic function $\operatorname{gap}\ge0$ with
$\operatorname{gap}(0)=0$ that is not identically $0$ starts with a term of even order $2j$ and positive coefficient.

\emph{No arc.} Suppose the interior $O$ of $\mathrm{Cont}$ in $\mathbb T$ is not empty. (i) Every boundary point $\zeta_0$ of $O$ in
$\mathbb T$ is in $\mathrm{Crit}$ and is isolated in $\mathbb T\setminus O$: take $\zeta_n\in O$, $\zeta_n\to\zeta_0$, with partners
$\zeta_n'\to\zeta_0'$. If $\zeta_0'\neq\zeta_0$, then $\zeta_n'=\tau(\zeta_n)$, so $\operatorname{gap}\equiv0$ at $\zeta_0$ and an arc about $\zeta_0$ lies in
$\mathrm{Cont}$, against $\zeta_0\notin O$. So $\zeta_0'=\zeta_0$, and $\zeta_0\in\mathrm{Crit}$ (else $W$ is injective near $\zeta_0$), $\zeta_n'=\beta(\zeta_n)$, $\operatorname{gap}\equiv0$ at $\zeta_0$, and a
punctured arc about $\zeta_0$ lies in $\mathrm{Cont}$, hence in $O$. Call such points tips; they lie in the finite set $\mathrm{Crit}$. (ii) $\mathbb T\setminus O$ consists of tips: otherwise a component arc of the non-empty open set
$\mathbb T\setminus\overline O\ne\mathbb T$ has an endpoint; it is a boundary point of $O$, hence a tip with a punctured neighbourhood
in $O$, while it is a limit of points of that arc, which are not in $O$.
(iii) Let $X=\mathbb C\setminus\overline{W(\mathbb D)}$, a non-empty open set since $W(\mathbb D)$ is bounded. Its boundary lies in
$\overline{W(\mathbb D)}\setminus W(\mathbb D)\subset W(\mathbb T)$. Let $\zeta_1\in O$ with partner $\zeta_2$. The partners of points
of $O$ near $\zeta_1$ tend to $\zeta_2$ (their limits are partners of $\zeta_1$, by (c)), so they are $\tau$ of them, so $|\tau|=1$ on an arc of $\mathbb T$ about $\zeta_1$. Replace $\Delta_1$ by a smaller
neighbourhood of $\zeta_1$, invariant under $w\mapsto1/\bar w$, such that $|\tau|=1$ on the arc $\mathbb T\cap\Delta_1$. By the reflection
principle, $\tau(1/\bar w)=1/\overline{\tau(w)}$ on $\Delta_1$ (both sides are holomorphic there and agree on that arc). For
$w\in\mathbb D\cap\Delta_1$ we have $W(\tau(w))=W(w)$ and $\tau(w)\neq w$, so $\tau(w)\notin\mathbb D$ by injectivity and
$\tau(w)\notin\mathbb T$ by (a): $|\tau(w)|>1$. Hence $|\tau|<1$ on $\Delta_1\setminus\overline{\mathbb D}$, so $\tau$ maps
$\Delta_1\setminus\overline{\mathbb D}$ into $\mathbb D$, and, as $W\circ\tau=W$, $W(\Delta_1)\subset W(\mathbb D)\cup W(\mathbb T\cap\Delta_1)$. As $W(\mathbb T\cap\Delta_1)$ is an analytic arc, $W(\Delta_1)$ does not meet $X$, so $W(\zeta_1)$ is not on the boundary
of $X$. Hence the boundary of $X$ lies in the finite set $W(\mathbb T\setminus O)$. As $\mathbb C$ minus a finite set is connected and
$X\neq\emptyset$, $X$ is the complement of a finite set, which contradicts $W(\mathbb D)\cap X=\emptyset$.
\end{proof}

\begin{theorem}\label{thm:S}
Let $(W,B)$ be an admissible pair with $\deg B=k\ge1$, and assume that $W$ extends holomorphically to a neighbourhood of
the closed disk. Then for every small $\delta>0$ there is a pair $(W_\delta,B_\delta)$ of the same kind with $\deg B_\delta=k$,
$B_\delta(0)=(1-\delta)B(0)$, and $W_\delta$ univalent on a neighbourhood of the closed disk. In particular $1/B(0)<\mathcal B_k$.
\end{theorem}

\begin{corollary}\label{cor:S}
Let $d\ge1$. If $(W,B)$ attains $\mathcal B_d$, then $W$ does not extend holomorphically to any neighbourhood of the closed disk.
\end{corollary}

\begin{corollary}\label{cor:S2}
If $W$ is rational without poles on the closed disk, then $(W,B)$ attains $\mathcal B_d$ for no $d\ge1$. This applies to every pair
of~\cite[Lemma~6.2]{JatarNg2026b} and of its extension, Lemma~\ref{lem:kzero}: there $W=H(\varphi)/N_b$, with $H$ a polynomial and $b$, $\varphi$ and $N_b$ as recalled at the
start of Section~\ref{sec:onezero}, is a polynomial if $b=0$ and otherwise has the only
pole $-1/\bar b$. So it applies to all pairs of Section~\ref{sec:onezero} and to the pairs of Proposition~\ref{prop:KLC}.
\end{corollary}

The corollaries follow from Theorem~\ref{thm:S} as Corollary~\ref{cor:ES} follows from Lemma~\ref{lem:ES}.

\begin{proof}[Proof of Theorem~\ref{thm:S}]
Let $F=WB$, let $w_1,\dots,w_p$ be the zeros of $F'$ on $\mathbb T$, of orders $m_i$, and let $U_1$, $B_\delta$ be as in the proof of
Lemma~\ref{lem:ES}. By Lemma~\ref{lem:D} let $\mathrm{Crit}=\{c_1,\dots,c_q\}$ and let $\{\zeta_l,\zeta_l'\}$ ($1\le l\le r$) be the contact
pairs. At $c\in\mathrm{Crit}$, $F'(c)=W(c)B'(c)\ne0$, since $W(c)\ne0$ by Lemma~\ref{lem:D}(a) and $cB'(c)/B(c)>0$; so $0$, the $w_i$
and the $c_j$ are distinct, and the contact points are distinct from $0$, from the $c_j$ and from each other. A contact point may equal
some $w_i$; let $e$ be the number of such points. Fix pairwise disjoint small disks about the $c_j$ and the contact points.

The plan. We add to $W$ a small multiple $\delta\upsilon_\delta$ of a polynomial, chosen by Hermite interpolation so that, as in
Lemma~\ref{lem:ES}, $(W_\delta B_\delta)'$ has no zero in $\mathbb D$, and so that every defect moves outward: the zero of $W_\delta'$
near a point $c$ of $\mathrm{Crit}$ is $(1+\delta)c+O(\delta^2)$, and for a contact pair $(\zeta_1,\zeta_2)$ the point near $\zeta_2$
with the value $W_\delta(\zeta_1)$ is $(1+\delta)\zeta_2+O(\delta^2)$; both lie outside the closed disk. The step `Local injectivity'
then shows that $W_\delta$ is injective on the closed disk near each defect, and `Global injectivity' that it is univalent near the
closed disk.

\emph{The polynomial $\upsilon_\delta$.} Impose: $\upsilon(0)=\upsilon'(0)=0$; at each $w_i$ the conditions of Lemma~\ref{lem:ES} for the
derivatives of orders $1,\dots,m_i$ of $\upsilon B_\delta$, with $\upsilon(w_i)=0$ unless $w_i$ is a contact point; at each $c_j$,
$\upsilon(c_j)=0$ and $\upsilon'(c_j)=-c_jW''(c_j)$; at each pair, $\upsilon(\zeta_l')=0$ and $\upsilon(\zeta_l)=\zeta_l'W'(\zeta_l')$. The
conditions at $w_i$ only make $\mathcal E_\delta$ vanish to order $m_i$ at $w_i$, whatever $\upsilon(w_i)$ is, and given $\upsilon(w_i)$ they form a
triangular system for $\upsilon^{(1)}(w_i),\dots,\upsilon^{(m_i)}(w_i)$ with diagonal $B_\delta(w_i)$. So all conditions are Hermite
conditions at distinct nodes, $n_H$ in number, where $n_H=2+\sum_i(m_i+1)+2q+2r-e$, and we let $\upsilon_\delta$ be the unique interpolant of
degree at most $n_H-1$. Its data are bounded uniformly in $\delta$, so $\upsilon_\delta$ is bounded in every $C^m(U_1)$ uniformly in $\delta$.
Put $W_\delta=W+\delta\upsilon_\delta$, $F_\delta=W_\delta B_\delta$. As in Lemma~\ref{lem:ES}, $F_\delta'\neq0$ in $\mathbb D$ for small
$\delta$, $W_\delta(0)=0$ and $W_\delta'(0)=1$; it remains to show that $W_\delta$ is univalent near the closed disk. Below, $C$ is a
constant independent of $\delta$.

\emph{The defects (the points of $\mathrm{Crit}$ and the contact pairs) move apart.} At $c=c_j$, $W_\delta'$ has one zero $c_\delta=c-\delta\upsilon_\delta'(c)/W''(c)+O(\delta^2)
=(1+\delta)c+O(\delta^2)$ near $c$. Write $W_\delta=W_\delta(c_\delta)+\hat x_\delta^2$. As $W_\delta-W=\delta\upsilon_\delta$ with $\upsilon_\delta$ bounded in $C^m$, and
$c_\delta-c=O(\delta)$, the square-root coordinate $\hat x_\delta$ and its inverse move by $O(\delta)$ in $C^1$: $|\hat x_\delta-\hat x|_{C^1}\le C\delta$ on a fixed disk,
and $\beta_\delta=\hat x_\delta^{-1}(-\hat x_\delta)$ has $|\beta_\delta-\beta|_{C^1}\le C\delta$ and $|\beta_\delta''|\le C$ on a fixed smaller
disk; as $\beta_\delta(c_\delta)=c_\delta$ and $\beta_\delta'(c_\delta)=-1$, $\log|\beta_\delta(c)|=2\delta+O(\delta^2)$. At a pair
$(\zeta_1,\zeta_2)=(\zeta_l,\zeta_l')$, $\tau_\delta=(W_\delta|_{\Delta_2})^{-1}\circ W_\delta$ has $|\tau_\delta-\tau|_{C^1}\le C\delta$ (likewise: the local
inverse of $W_\delta$ near $\zeta_2$, where $W'\neq0$, moves by $O(\delta)$ in $C^1$) and
$\tau_\delta(\zeta_1)-\zeta_2=\delta(\upsilon_\delta(\zeta_1)-\upsilon_\delta(\zeta_2))/W'(\zeta_2)+O(\delta^2)=\delta\zeta_2+O(\delta^2)$, so
$\log|\tau_\delta(\zeta_1)|=\delta+O(\delta^2)$.

\emph{Local injectivity.} Near a point $c$ of $\mathrm{Crit}$, the only point $w'\ne w$ near $c$ with $W_\delta(w')=W_\delta(w)$ is
$\beta_\delta(w)$; near a contact pair $(\zeta_1,\zeta_2)$, the point near $\zeta_2$ with the value $W_\delta(w)$ of a point $w$
near $\zeta_1$ is $\tau_\delta(w)$. So it suffices to show that $|\beta_\delta|>1$ on the closed disk near $c$ and $|\tau_\delta|>1$
on the closed disk near $\zeta_1$: then these points lie outside the closed disk. Let $\phi=\beta$ at $\zeta_0=\zeta_0^*=c$, or $\phi=\tau$ at $\zeta_0=\zeta_1$, $\zeta_0^*=\zeta_2$, and
$G_\delta(\rho,s)=\log|\phi_\delta(\rho\zeta_0e^{is})|$ on the box $1-r_b\le\rho\le1$, $|s|\le r_b$, with $r_b$ fixed. At $\delta=0$,
$G(1,s)=\operatorname{gap}(s)\ge c_{\mathrm{gap}} s^{2j}$ by Lemma~\ref{lem:D}(e), and $\partial_\rho G(1,0)=\mathrm{Re}(\bar\zeta_0^*\phi'(\zeta_0)\zeta_0)=-\alpha_0<0$: for $\beta$, $\alpha_0=1$;
for $\tau$, the number $\bar\zeta_2\tau'(\zeta_1)\zeta_1$ is real since $\operatorname{gap}'(0)=0$, it is not positive since $|\tau|\ge1$ inside the disk and
$|\tau(\zeta_1)|=1$, and it is not $0$, since $\tau'(\zeta_1)=W'(\zeta_1)/W'(\zeta_2)\neq0$, so $\alpha_0=|\tau'(\zeta_1)|$. Hence $G(\rho,s)\ge c_{\mathrm{gap}} s^{2j}+(\alpha_0/2)(1-\rho)$ on the box for small
$r_b$. Since $|\nabla(G_\delta-G)|\le C\delta$, $G(1,0)=0$ and $G_\delta(1,0)\ge\varpi\delta-C\delta^2$, where $\varpi=2$ at a point of $\mathrm{Crit}$ and $\varpi=1$ at a contact pair (the separations computed above), we have
$G_\delta(\rho,s)\ge G_\delta(1,0)+G(\rho,s)-C\delta(|s|+1-\rho)$, so
\[
  G_\delta(\rho,s)\ \ge\ \bigl(c_{\mathrm{gap}} s^{2j}-C\delta|s|\bigr)+(\alpha_0/2-C\delta)(1-\rho)+\varpi\delta-C\delta^2
  \ \ge\ \varpi\delta-C'\delta^{2j/(2j-1)}-C\delta^2\ >\ 0
\]
for small $\delta$: the first bracket is at least its minimum over $s$, which is $-C'\delta^{2j/(2j-1)}$ with
$C'=(1-\frac1{2j})\,C\,(C/(2jc_{\mathrm{gap}}))^{1/(2j-1)}$, attained at $|s|^{2j-1}=C\delta/(2jc_{\mathrm{gap}})$; the second term is non-negative once
$C\delta\le\alpha_0/2$; and $2j/(2j-1)>1$. So $|\beta_\delta|>1$ (resp. $|\tau_\delta|>1$) on the closed disk near $c$ (resp. $\zeta_1$), including the points of
$\mathbb T$. If $w\ne w'$ in the closed disk near $c$ have $W_\delta(w)=W_\delta(w')$, then $w'=\beta_\delta(w)$ lies outside the closed
disk; likewise near a pair, with $\tau_\delta$. So $W_\delta$ is injective on the closed disk near each defect, and $W_\delta'\ne0$ there,
its only zero near $c$ being $c_\delta\notin\overline{\mathbb D}$. Near any other point of the closed disk $W'\ne0$, and $W_\delta$ is
injective on a fixed disk about it by the segment argument of Lemma~\ref{lem:ES}.

\emph{Global injectivity.} If $\delta_n\to0$ and $w_n\ne w_n'$ in the closed disk had $W_{\delta_n}(w_n)=W_{\delta_n}(w_n')$, a subsequence
converges to $(w^*,w'^*)$ with $W(w^*)=W(w'^*)$. If $w^*=w'^*$ the local injectivity is contradicted; otherwise both points lie on
$\mathbb T$ by Lemma~\ref{lem:D}(a), so they form a contact pair, and the local injectivity at that pair is contradicted. So for small
$\delta$, $W_\delta$ is injective on the closed disk with $W_\delta'\ne0$ there, hence univalent on a neighbourhood of it.
\end{proof}

If $W$ is rational, as in Corollary~\ref{cor:S2}, then $W(\mathbb D)$ is a bounded quadrature domain~\cite{AharonovShapiro1976}, and
Lemma~\ref{lem:D} also follows from the known structure of the boundaries of such domains (finitely many analytic arcs with finitely many
cusps and double points); compare Sakai~\cite{Sakai1991} and Gustafsson~\cite{Gustafsson1983}. Schiffer's
boundary variation \cite[Ch.~10]{Duren1983} shows that extremals of differentiable functionals on $\mathcal S$ map onto the plane minus
analytic arcs; it does not give Theorem~\ref{thm:S}, since the pairs here are not closed under Schiffer's variations, and the Hermite
repair of Lemma~\ref{lem:ES} is what keeps $(WB)'\neq0$ in $\mathbb D$.

\medskip\noindent\emph{Poles on the circle.} We now allow poles on the circle; then $W(\mathbb D)$ is unbounded, as for the Koebe
function. Let $W\in\mathcal S$ be meromorphic on an open set $U\supset\overline{\mathbb D}$; shrinking $U$, its poles in $U$ lie on
$\mathbb T$ and form a finite set $P$. Define $\mathrm{Crit}$ and $\mathrm{Cont}$ as above with $\mathbb T\setminus P$ in place of
$\mathbb T$. A \emph{slit plane}\label{term:slit} is a domain $\mathbb C\setminus\Gamma$ with $\Gamma$ a non-empty finite union of analytic arcs, closed in
$\mathbb C$.

\begin{lemma}\label{lem:poles}
The orders of the poles of $W$ add up to at most $2$, and $W(\mathbb T\setminus P)\cap W(\mathbb D)=\emptyset$.
\end{lemma}

\begin{proof}
Near a pole $p$ of order $m$, $V=1/W=\hat x^m$ with $\hat x$ conformal at $p$, $\hat x(p)=0$, and $V$ is injective on $\mathbb D\cap\Delta$. As in
Lemma~\ref{lem:D}(b), $\hat x(\mathbb D\cap\Delta)$ contains a sector at $0$ of opening $\pi-\varepsilon$, and $u\mapsto u^m$ is injective on it;
so $m(\pi-\varepsilon)\le2\pi$ and $m\le2$. Then $V(\mathbb D\cap\Delta_p)$ contains a sector of opening $m(\pi-\varepsilon)$. These sets are
disjoint for different poles, so $\sum_pm_p(\pi-\varepsilon)\le2\pi$ for every
$\varepsilon>0$. The second statement is proved as Lemma~\ref{lem:D}(a).
\end{proof}

At a double pole $p$ write $1/W=\hat x^2$ with $\hat x$ conformal, and put $\beta_\infty=\hat x^{-1}(-\hat x)$. At two simple poles $p_1,p_2$ put
$\tau_\infty=(V|_{\Delta_2})^{-1}\circ V$ near $p_1$, so $\tau_\infty(p_1)=p_2$. A lone simple pole has no partner, as $V$ is injective
near it. As in the proof of Lemma~\ref{lem:D}, $\operatorname{gap}_\infty(s)=\log|\beta_\infty(pe^{is})|$, resp. $\log|\tau_\infty(p_1e^{is})|$, is
real-analytic near $0$, $\operatorname{gap}_\infty\ge0$ and $\operatorname{gap}_\infty(0)=0$.

\begin{lemma}\label{lem:Dplus}
Either $\mathrm{Cont}$ is finite, and every gap function of Lemma~\ref{lem:D}\textup{(e)} (defined in the same way on $\mathbb T\setminus P$)
and every $\operatorname{gap}_\infty$ is at least $c_{\mathrm{gap}} s^{2j}$ near $0$, for some $c_{\mathrm{gap}}>0$ and $j\ge1$; or $W(\mathbb D)=\mathbb C\setminus\Gamma$ is a slit plane with
$\Gamma=W(\mathbb T\setminus P)$, all but finitely many points of $\Gamma$ have exactly two preimages in $\mathbb T\setminus P$, every
component of $\Gamma$ is unbounded, and $P$ consists of one double pole or of two simple poles.
\end{lemma}

\begin{proof}
The proof has two parts. First, if $\mathrm{Cont}$ is infinite, or some $\operatorname{gap}$ or $\operatorname{gap}_\infty$ vanishes identically, then
$\mathrm{Cont}$ contains an arc. Second, if $\mathrm{Cont}$ contains an arc, let $O$ be its interior: we show in turn that $\mathbb
T\setminus O$ is a finite set of tips, that $\Gamma=\mathbb C\setminus W(\mathbb D)$ equals $W(\mathbb T\setminus P)$, and that the
partner map pairs the arcs of $O$, so that all but finitely many points of $\Gamma$ have exactly two preimages and every component of
$\Gamma$ is unbounded.

Parts (b), (c) of Lemma~\ref{lem:D} and the local maps $\beta,\tau$ are local at finite values, so they hold on $\mathbb T\setminus P$.
$\mathrm{Crit}$ is finite, as $|W'|\to\infty$ at the poles, so the zeros of $W'$ on $\mathbb T\setminus P$ do not accumulate.
If $\mathrm{Cont}$ is infinite, or some $\operatorname{gap}$ or $\operatorname{gap}_\infty$ vanishes identically, then $\mathrm{Cont}$ contains an arc. For contact pairs
$(\zeta_n,\zeta_n')$ with bounded values this is the reduction in the proof of Lemma~\ref{lem:D}. If the values tend to $\infty$, both
points tend to poles: to one double pole, with $\zeta_n'=\beta_\infty(\zeta_n)$, or to the two simple poles, with
$\zeta_n'=\tau_\infty(\zeta_n)$ up to order (a lone simple pole is excluded, as $V$ is injective near it); then $\operatorname{gap}_\infty\equiv0$, and
$\beta_\infty$ or $\tau_\infty$ maps an arc of $\mathbb T$ into $\mathbb T$.

Let $O\ne\emptyset$ be the interior of $\mathrm{Cont}$ in $\mathbb T$. Every boundary point of $O$ is a tip, that is, isolated in
$\mathbb T\setminus O$: for a point not in $P$ as in Lemma~\ref{lem:D} (the limit partner has a finite value, so it is not a pole); at a
double pole, or at one of two simple poles, the partners of nearby points of $O$ tend to a pole, so $\operatorname{gap}_\infty\equiv0$ and a punctured arc
about the pole lies in $O$. A lone simple pole is not a boundary point of $O$, since a neighbourhood of it in $\mathbb T$ misses
$\mathrm{Cont}$. As in Lemma~\ref{lem:D}, $\mathbb T\setminus O$ consists of finitely many tips; in particular there is no lone simple pole.
With $X=\mathbb C\setminus\overline{W(\mathbb D)}$, the proof of Lemma~\ref{lem:D} shows that the boundary of $X$ lies in the finite set
$W(\text{tips}\setminus P)$ (the step (iii) is local at a finite value), so $X=\emptyset$: a non-empty open set with finite boundary is
the complement of a finite set, and it would meet $W(\mathbb D)$; here $\overline{W(\mathbb D)}\setminus W(\mathbb D)\subset W(\mathbb T\setminus P)$, since a finite limit
of values $W(w_n)$ comes from points $w_n$ that tend to a point of the closed disk other than a pole. Hence $\Gamma:=\mathbb C\setminus W(\mathbb D)=\overline{W(\mathbb D)}\setminus W(\mathbb D)
=W(\mathbb T\setminus P)$, by Lemma~\ref{lem:poles}, and $\Gamma=W(O)\cup W(\text{tips}\setminus P)$. As $W(\mathbb D)$ is dense, it is unbounded, so $P\neq\emptyset$;
since poles are not in $\mathrm{Cont}$, $O$ lies in $\mathbb T\setminus P\neq\mathbb T$, and every component of $O$ is an open arc. The partner of a point of $O$
lies in $\mathrm{Cont}$ and is not a tip (a tip in $\mathrm{Crit}$ has no partner, and a pole is not in $\mathbb T\setminus P$), so it lies in
$O$, and the partner map is a continuous involution of $O$ without fixed points, as $\mathrm{Crit}\cap\mathrm{Cont}=\emptyset$; since an open interval has no such involution (a continuous involution of an
interval is monotone; an increasing one is the identity, and a decreasing one has a fixed point), it
maps each component arc $A$ of $O$ onto another one, $A'$. So $W$ is injective on $A$, and $W(A)=W(A')$ is an analytic arc whose points
have exactly two preimages. $\Gamma$ is not empty, since $W(\mathbb D)$ misses $W(\mathbb T\setminus P)$. By Lemma~\ref{lem:poles}, $P$ is one double pole or two simple poles. Finally $\Gamma\cup\{\infty\}$,
the complement of the simply connected domain $W(\mathbb D)$ in the sphere, is connected, and $\Gamma$ has finitely many components; a
bounded one would be compact and would split $\Gamma\cup\{\infty\}$.
\end{proof}

\begin{theorem}\label{thm:Splus}
Let $(W,B)$ be an admissible pair with $\deg B=k\ge1$, let $W$ be meromorphic on a neighbourhood of the closed disk, and let
$\mathrm{Cont}$ be finite. Then for every small $\delta>0$ there is a pair $(W_\delta,B_\delta)$ of the same kind with $\deg B_\delta=k$
and $B_\delta(0)=(1-\delta)B(0)$. In particular $1/B(0)<\mathcal B_k$.
\end{theorem}

\begin{corollary}\label{cor:Splus}
Let $d\ge1$. If $(W,B)$ attains $\mathcal B_d$ and $W$ extends meromorphically to a neighbourhood of the closed disk, then $W(\mathbb D)$
is the plane minus a non-empty finite union $\Gamma$ of analytic arcs, each traced twice by $W$ on the circle, every component of $\Gamma$
is unbounded, and $W$ has a double pole or two simple poles on the circle.
\end{corollary}

For the corollary, Theorem~\ref{thm:Splus} shows that $\mathrm{Cont}$ is infinite (as in the proof of Corollary~\ref{cor:ES}),
and Lemma~\ref{lem:Dplus} gives the rest.

\begin{proof}[Proof of Theorem~\ref{thm:Splus}]
We follow the proof of Theorem~\ref{thm:S}; near the points of $\mathbb T\setminus P$ nothing changes. To the conditions on $\upsilon_\delta$
we add, at each pole $p_\infty$, $\upsilon^{(l)}(p_\infty)=0$ for $0\le l<2j$, where $j$ is the exponent of $\operatorname{gap}_\infty$ in Lemma~\ref{lem:Dplus}
($\operatorname{gap}_\infty(s)\ge c_{\mathrm{gap}} s^{2j}$) at a double pole, resp.\ at $p_1$ for two simple poles, and $j=0$ for a lone simple pole. The poles are
distinct from the other nodes, and $W_\delta=W+\delta\upsilon_\delta$ has the same poles with the same orders.

\emph{$F_\delta'\neq0$ in $\mathbb D$.} Near a pole $p_\infty$ of order $m$, with $R_\delta=(B_\delta-B)/\delta$,
\[
  \frac{\mathcal E_\delta}{F'}=\frac{R_\delta+(W/W')R_\delta'+(\upsilon_\delta B_\delta)'/W'}{B+(W/W')B'},
\]
where $W/W'=O(|w-p_\infty|)$ and $1/W'=O(|w-p_\infty|^{m+1})$. As $B(p_\infty)\ne0$, $\mathcal E_\delta/F'$ is bounded on a fixed disk about $p_\infty$ uniformly in $\delta$;
on the rest of the closed disk the proof of Lemma~\ref{lem:ES} applies. So $F_\delta'\neq0$ in $\mathbb D$ for small $\delta$.

\emph{Injectivity near the poles.} Put $V_\delta=1/W_\delta=Vu_\delta$ with $u_\delta=1/(1+\delta\upsilon_\delta V)$. On $|w-p_\infty|\le r$,
$|u_\delta-1|\le C\delta r^{2j+m}$ and $|u_\delta'|\le C\delta r^{2j+m-1}$. At a simple pole, $V'(p_\infty)\neq0$; fix a disk of radius at most $r$ about $p_\infty$ on which
$|V'-V'(p_\infty)|<|V'(p_\infty)|/4$. For small $\delta$, $|V_\delta'-V'(p_\infty)|<|V'(p_\infty)|/2$ there, and integration along the
segment from $w'$ to $w$ gives $|V_\delta(w)-V_\delta(w')|\ge|V'(p_\infty)|\,|w-w'|/2>0$ for $w\neq w'$ in it, as in the proof of
Lemma~\ref{lem:ES}; so $V_\delta$ is injective there. At a double pole, $\hat x_\delta=\hat x u_\delta^{1/2}$ is conformal with
$|\hat x_\delta-\hat x|_{C^1}\le C\delta r^{2j+1}$, and $\beta_{\infty,\delta}=\hat x_\delta^{-1}(-\hat x_\delta)$ fixes $p_\infty$. At two simple poles
$\tau_{\infty,\delta}(p_1)=p_2$, and $V(\tau_{\infty,\delta})u_\delta(\tau_{\infty,\delta})=Vu_\delta$ gives
$|\tau_{\infty,\delta}-\tau_\infty|_{C^1}\le C\delta r^{2j}$, since by the bounds on $u_\delta$ near $p_1$ and near $p_2$ the map $V_\delta$ and its
local inverse near $p_2$ differ from $V$ and its inverse by $O(\delta r^{2j})$ in $C^1$ (this uses the condition at both poles). With
$G_\delta(\rho,s)=\log|\phi_\delta(\rho\zeta_0e^{is})|$ for $\phi=\beta_\infty$ or $\tau_\infty$, as in the proof of Theorem~\ref{thm:S}, we get
$G_\delta(1,0)=G(1,0)=0$. The maps $\beta_{\infty,\delta}$ and $\beta_\infty$ fix $p_\infty$, and the bound on $\hat x_\delta$ gives
$|\beta_{\infty,\delta}-\beta_\infty|_{C^1}\le C\delta r^{2j+1}$ on $|w-p_\infty|\le r$; integrating from $p_\infty$ along a segment gives
$|\beta_{\infty,\delta}(w)-\beta_\infty(w)|\le C\delta|w-p_\infty|^{2j+2}$. Likewise $\tau_{\infty,\delta}(p_1)=\tau_\infty(p_1)=p_2$ gives
$|\tau_{\infty,\delta}(w)-\tau_\infty(w)|\le C\delta|w-p_1|^{2j+1}$. As $|w-\zeta_0|\le C(|s|+1-\rho)$ at $w=\rho\zeta_0e^{is}$, and $\log|\cdot|$ is
Lipschitz near the circle, $|G_\delta-G|\le C\delta(|s|+1-\rho)^{2j+1}$: the exponent is $2j+2$ or $2j+1$, and $|s|+1-\rho\le2$, since $|s|\le1$ and $0\le1-\rho\le1$ on the box. So
$(|s|+1-\rho)^{2j+1}\le2^{2j+1}\bigl(|s|^{2j+1}+(1-\rho)^{2j+1}\bigr)\le2^{2j+1}\bigl(s^{2j}+(1-\rho)\bigr)$. Since
$G(\rho,s)\ge c_{\mathrm{gap}} s^{2j}+(\alpha_0/2)(1-\rho)$ on a small box,
\[
  G_\delta(\rho,s)\ \ge\ (c_{\mathrm{gap}}-C'\delta)s^{2j}+(\alpha_0/2-C'\delta)(1-\rho)\ >\ 0\qquad\text{for }(\rho,s)\ne(1,0)
\]
and small $\delta$. So $W_\delta$ is injective on the closed disk minus $P$ near each pole.

\emph{Global injectivity.} If $W_{\delta_n}(w_n)=W_{\delta_n}(w_n')\to\infty$ with $w_n\ne w_n'$ in $\mathbb D$, then $w_n,w_n'$ tend to
poles, against the previous step; bounded values are treated as in the proof of Theorem~\ref{thm:S}. So $W_\delta\in\mathcal S$.
\end{proof}

Theorem~\ref{thm:Splus} does not say that $W_\delta$ is univalent past the circle; near a double pole it cannot be.
Whether an extremal $W$ is meromorphic past the circle is open. For
unbounded quadrature domains compare Shapiro~\cite{Shapiro1987} and Sakai~\cite{Sakai1991}.