\subsection{Pairs that are univalent past the circle are not extremal}\label{sec:ES}

From here on we write $F$ for $WB$; so $F=\Phi\circ W$, with $\Phi(z)=zB(W^{-1}(z))$ on $W(\mathbb D)$ as in
Section~\ref{sec:structure}. Recall that, for $d\ge0$, $\mathcal B_d$ is the supremum of $1/B(0)$ over the pairs $(W,B)$ with $W\in\mathcal S$, $B$ a finite Blaschke product of
degree at most $d$ with $B(0)>0$, and $(WB)'\neq0$ on $\mathbb D$; by~\cite[Thm.~8.9]{JatarNg2026b} each $\mathcal B_d$ is attained. The
zeros of such a $B$ are simple and non-zero, since a multiple zero, or a zero at $0$, would be a zero of $(WB)'$ in $\mathbb D$.

\begin{lemma}\label{lem:ES}
Let $(W,B)$ be as above with $\deg B=k\ge1$, and assume that $W$ extends to a univalent function on 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 $W_\delta$ again univalent on a neighbourhood of the closed disk. In particular $1/B(0)<\mathcal B_k$.
\end{lemma}

\begin{proof}
Let $W$ be univalent on an open set $U\supset\overline{\mathbb D}$, put $F=WB$, and let $B_\delta$ be $B$ with one zero $a_1$ replaced by
$(1-\delta)a_1$; the normalising constant depends on $\arg a_1$ only, so $B_\delta(0)=(1-\delta)B(0)$. Since $F'(0)=B(0)\neq0$, $F'$ has
finitely many zeros $w_1,\dots,w_p$ on the circle, of orders $m_1,\dots,m_p$, and none in $\mathbb D$. Fix a compact neighbourhood $U_1\subset U$
of the closed disk, free of the poles of $B$ and $B_\delta$, on which the $w_i$ are the only zeros of $F'$. Let $\upsilon_\delta$ be the
polynomial of degree at most $1+\sum_i(m_i+1)$ with $\upsilon_\delta(0)=\upsilon_\delta'(0)=0$, $\upsilon_\delta(w_i)=0$ and
\[
  (\upsilon_\delta B_\delta)^{(j)}(w_i)=-\bigl(W(B_\delta-B)/\delta\bigr)^{(j)}(w_i)\qquad(1\le j\le m_i,\ 1\le i\le p).
\]
By Leibniz's rule these conditions form a triangular system for the jet of $\upsilon_\delta$ at $w_i$ with diagonal $B_\delta(w_i)$, of
modulus $1$; so $\upsilon_\delta$ exists, depends linearly on the data, and, as $(B_\delta-B)/\delta$ is bounded with its derivatives on
$U_1$ uniformly in $\delta$, it is bounded with its derivatives on $U_1$ uniformly in $\delta$. Put $W_\delta=W+\delta\upsilon_\delta$ and
$F_\delta=W_\delta B_\delta$. Then $F_\delta'-F'=\delta {\mathcal E}_\delta$, where $\mathcal E_\delta$ is the derivative of $W(B_\delta-B)/\delta+\upsilon_\delta
B_\delta$; it vanishes to order at least $m_i$ at each $w_i$ and is bounded on $U_1$ uniformly in $\delta$. Hence $\mathcal E_\delta/F'$ is
holomorphic near the closed disk and, by the maximum principle on small fixed circles about the $w_i$ together with a positive lower
bound for $|F'|$ elsewhere on $\overline{\mathbb D}$, bounded there by some $C_1$ independent of $\delta$. So
$F_\delta'=F'(1+\delta {\mathcal E}_\delta/F')$ has no zero in $\mathbb D$ once $\delta {C_1}<1$. Finally $W_\delta(0)=0$, $W_\delta'(0)=1$, and
$W_\delta$ is injective on a closed disk $E$ of radius $R>1$ inside $U_1$ for small $\delta$. Indeed, let $|W'|\ge c_W>0$ on $E$, let $C_2$
and $C_3$ bound $|\upsilon_\delta'|$ and $|\upsilon_\delta|$ on $E$ uniformly in $\delta$, and choose $\eta>0$ with $|W'(\zeta)-W'(\zeta')|<c_W/2$
whenever $\zeta,\zeta'\in E$ and $|\zeta-\zeta'|<\eta$. Let $w\neq w'$ in $E$. If $|w-w'|<\eta$, integration along the segment from $w'$
to $w$ gives $|W(w)-W(w')|\ge(c_W/2)|w-w'|>\delta {C_2}|w-w'|\ge\delta|\upsilon_\delta(w)-\upsilon_\delta(w')|$ once $\delta {C_2}<c_W/2$. If
$|w-w'|\ge\eta$, then $|W(w)-W(w')|\ge c(\eta)$ for some $c(\eta)>0$, since $W$ is injective on the compact set $E$, while
$\delta|\upsilon_\delta(w)-\upsilon_\delta(w')|\le2\delta {C_3}<c(\eta)$ for small $\delta$. In both cases $W_\delta(w)\neq W_\delta(w')$.
\end{proof}

If $F'\neq0$ on the closed disk, no repair is needed ($\upsilon_\delta=0$); the content of Lemma~\ref{lem:ES} is the repair at the
zeros of $F'$ on the circle.

\begin{corollary}\label{cor:ES}
Let $d\ge1$. If $(W,B)$ attains $\mathcal B_d$, then $W$ does not extend to a univalent function on any neighbourhood of the closed
disk. In particular no pair attaining $\mathcal B_d$ is a Blaschke model.
\end{corollary}

\begin{proof}
$\mathcal B_d\ge\mathcal B_1>1$~\cite[Thm.~6.1]{JatarNg2026b}, so $k=\deg B\ge1$, and $\mathcal B_k\le\mathcal B_d$. Lemma~\ref{lem:ES} would give
$\mathcal B_d=1/B(0)<\mathcal B_k\le\mathcal B_d$.
\end{proof}

Lemma~\ref{lem:ES} gives no explicit lower bound for $\mathcal B_k-1/B(0)$. Corollary~\ref{cor:ES} settles the point left open in~\cite[Rem.~8.11]{JatarNg2026b} (no extremal pair is regular), and shows that for $d\ge1$ the hypothesis of the last
sentence of~\cite[Prop.~8.8]{JatarNg2026b} (a regular pair attaining $\mathcal B_d$) is never met.

\begin{lemma}[{Schur--Cohn~\cite{Marden}; compare~\cite[Lemma~6.4]{JatarNg2026b}}]\label{lem:SC}
Let $p(w)=e_0+\dots+e_nw^n$ with $n\ge1$ ($e_n$ may vanish), $p^*(w)=w^n\overline{p(1/\bar w)}$ and $\mathcal Rp=\bar e_0\,p-e_n\,p^*$, a
polynomial of formal degree $n-1$ with constant term $|e_0|^2-|e_n|^2$. If $|e_0|>|e_n|$, then $p$ and $\mathcal Rp$ have the same zeros on the
circle, and if $p$ has none there, the same number of zeros in $\mathbb D$. Consequently, if $p_0=p$, $p_j=\mathcal Rp_{j-1}$ and the constant
terms $\delta_j$ of $p_j$ ($1\le j\le n$) are all positive, then $p$ has no zero in the closed disk.
\end{lemma}

\begin{proof}
On the circle $|p^*|=|p|$. If $p(w)=0$ there, then $\mathcal Rp(w)=0$; if $\mathcal Rp(w)=0$, then $|e_0||p(w)|=|e_n||p(w)|$, so $p(w)=0$. If $p$ has no
zero on the circle, then $|e_np^*|<|\bar e_0p|$ there, and Rouch\'e's theorem applies. For the last claim go back from the non-zero
constant $p_n=\delta_n$.
\end{proof}

Corollary~\ref{cor:ES2}, in Section~\ref{sec:onezero}, applies Lemma~\ref{lem:SC} to the two degree-one pairs
of~\cite[Thm.~6.1, Rem.~6.7]{JatarNg2026b}: their maps $W$ are univalent on a neighbourhood of the closed disk, so these
pairs are not extremal.