\subsection{An upper bound in degree two}\label{sec:B2upper}

We use the notation of Section~\ref{sec:B1upper}, with $|\operatorname{Log}\xi|$ the least modulus of a logarithm of
$\xi\neq0$, and we put $|\operatorname{Log}0|=+\infty$. Let $B(w)=\lambda\prod_{j=1}^{d}(w-a_j)/(1-\bar a_jw)$ be a Blaschke product of degree $d\ge1$ with
simple nonzero zeros, and put
\[
  \sigma=\frac{wB'}{B}=\sum_{j=1}^{d}\frac{(1-|a_j|^2)w}{(w-a_j)(1-\bar a_jw)}=\frac{\check N}{\check R},\qquad {\check R}=\prod_{j=1}^{d}(w-a_j)(1-\bar a_jw),
\]
\[
  \check N=\sum_{j=1}^{d}(1-|a_j|^2)\,w\prod_{l\neq j}(w-a_l)(1-\bar a_lw),\qquad \widehat P_t=\check Re^{t\ell}+\check N\quad(0\le t\le1).
\]
The proof of~\cite[Thm.~7.4]{JatarNg2026b} gives the following statement for any number of zeros and any circle in the disk.

\begin{lemma}\label{lem:homotopyd}
Let $\Delta=\{|w-w_C|\le\rho\}$ be a closed disk contained in $\mathbb D$ whose boundary circle $C$ contains no $a_j$, and assume
$|\operatorname{Log}(-\sigma(w))|>L(|w|)$ for every $w\in C$. If $wB$ has a critical point in the interior of $\Delta$, then for every $W\in\mathcal S$
the derivative $(WB)'$ has a zero in the interior of $\Delta$.
\end{lemma}

\begin{proof}
Zeros are counted with multiplicity. Since $wW'/W=e^{\ell}$ and $B/\check R=\lambda/\prod_j(1-\bar a_jw)^2$,
\[
  (WB)'=\frac Ww\,B\,(e^{\ell}+\sigma)=\frac{\lambda W}{w}\,\frac{\widehat P_1}{\prod_j(1-\bar a_jw)^2};
\]
here $W(w)/w$ extends holomorphically to $\mathbb D$ with value $1$ at $0$ and has no zero in $\mathbb D$, so the zeros of $(WB)'$ in
$\mathbb D$ are those of $\widehat P_1$. Next, $\widehat P_0=\check R(1+\sigma)$ and $(wB)'=B(1+\sigma)$, so $\widehat P_0=(wB)'\prod_j(1-\bar a_jw)^2/\lambda$, and the zeros
of $\widehat P_0$ in $\mathbb D$ are the critical points of $wB$. Let $w\in C$. Then $\check R(w)\neq0$, since $w\neq a_j$ and
$|1-\bar a_jw|\ge1-|w|>0$. If $\sigma(w)=0$, then $\widehat P_t(w)=\check R(w)e^{t\ell(w)}\neq0$. Otherwise $\widehat P_t(w)=0$ would mean $e^{t\ell(w)}=-\sigma(w)$,
so $t\ell(w)$ would be a logarithm of $-\sigma(w)$ and $|\operatorname{Log}(-\sigma(w))|\le t|\ell(w)|\le L(|w|)$ by~\eqref{eq:grunsky}, a
contradiction. Hence no $\widehat P_t$ vanishes on $C$. Since $\widehat P_t$ is jointly continuous in $(t,w)\in[0,1]\times\mathbb D$ and holomorphic in
$w$, and $\Delta\subset\mathbb D$, the number of zeros of $\widehat P_t$ inside $C$ does not depend on $t$, and it is positive at $t=0$.
\end{proof}

The disk must lie in $\mathbb D$: $\ell$ and~\eqref{eq:grunsky} are defined there only, and only the zeros of $\widehat P_0$ in $\mathbb D$ are
critical points of $wB$. A zero $a_j$ inside $C$ does no harm, since $\widehat P_0(a_j)=\check N(a_j)\neq0$, and a circle through $0$ is allowed,
since $\sigma(0)=0$.
For $w_C=0$ and $d=1$ the lemma is Lemma~\ref{lem:circlehomotopy}; unlike the circle
$|w|=s$ of~\cite[Thm.~7.4]{JatarNg2026b}, the circles used below keep away from every $a_j$.

\begin{theorem}\label{thm:B2upper}
Let $W\in\mathcal S$ and let $B$ be a Blaschke product of degree two with $B(0)>0$ such that $(WB)'$ has no zero in $\mathbb D$. Then
the zeros $a_1,a_2$ of $B$ satisfy $|a_1a_2|>61/100$. Consequently
\[
  \mathcal B_2<\frac{100}{61}=1.63934\ldots<1.639345 .
\]
\end{theorem}

\begin{proof}
The zeros of $B$ are simple and nonzero: $B(0)\neq0$, and at a multiple zero $a$ one has $(WB)'(a)=W'(a)B(a)+W(a)B'(a)=0$. A rotation
$W(w)\mapsto e^{-i\theta}W(e^{i\theta}w)$, $B(w)\mapsto B(e^{i\theta}w)$ (as in the proof of Theorem~\ref{thm:B1upper}) and the
conjugation $W(w)\mapsto\overline{W(\bar w)}$, $B(w)\mapsto\overline{B(\bar w)}$ keep the hypotheses and $|a_1a_2|=B(0)$. Rotating the
zero of larger modulus to the positive axis, we may assume $a_1=r_1\in(0,1)$, $a_2=x_2+iy_2$ with $y_2\ge0$, $0<|a_2|\le r_1$ and
$a_2\neq a_1$. Suppose $r_1|a_2|\le61/100$. The compact set
\[
  X_{61}=\{(r_1,x_2,y_2):0\le r_1\le1,\ x_2^2+y_2^2\le r_1^2,\ y_2\ge0,\ r_1\sqrt{x_2^2+y_2^2}\le61/100\}
\]
is covered by finitely many boxes, products of three closed intervals with rational ends, and each box comes with a rational point
$w_C$ and a rational $\rho>0$ such that, for every parameter in the box: (i) $|w_C|+\rho\le199/200$, $|a_2|<1$, and the circle
$C=\{|w-w_C|=\rho\}$ has distance more than $1/200$ from $a_1$ and from $a_2$; (ii) $|\operatorname{Log}(-\sigma(w))|>L(|w|)$ for every $w\in C$; and
(iii) at one rational point of the box, $\widehat P_0$ has $m\ge1$ zeros inside $C$. By (i), $\check R\neq0$ on $C$, since
$|w-a_j|>1/200$ and $|1-\bar a_jw|\ge1-|w|\ge1/200$, and by (ii), $1+\sigma\neq0$ on $C$, since $|\operatorname{Log}(-\sigma)|>L(|w|)\ge0=|\operatorname{Log}1|$; so $\widehat P_0=\check R(1+\sigma)$ has
no zero on $C$ for any parameter in the box. The coefficients of $\widehat P_0$ are polynomials in $(r_1,x_2,y_2)$, so by the argument principle
the number of zeros of $\widehat P_0$ inside $C$ is $m\ge1$ on the whole box; by (i) these zeros lie in $\mathbb D$, so they are critical points
of $wB$. Lemma~\ref{lem:homotopyd} with the circle of the box containing $(r_1,x_2,y_2)$ gives a zero of $(WB)'$ in $\mathbb D$, a
contradiction. Hence $|a_1a_2|>61/100$. The points of $X_{61}$ with $r_1=1$, $a_2=0$ or $a_2=a_1$ are not of this form; they only make $X_{61}$
compact, and all functions used are continuous there by (i). Finally, $1/B(0)=1/|a_1a_2|<100/61$, a pair of degree at most one has
gain less than $\sqrt{7/5}=1.18321\ldots<100/61$ by~\cite[Thm.~7.4, Cor.~8.10]{JatarNg2026b}, and $\mathcal B_2$ is attained~\cite[Thm.~8.9]{JatarNg2026b};
so $\mathcal B_2<100/61$.
\end{proof}

The computation that gives the boxes and checks (i)--(iii) is described in Appendix~\ref{app:computations}.