\section{Open problems}\label{sec:open}

\begin{enumerate}
\item \emph{Do more zeros help?} Jatar and Ng~\cite[Sect.~9]{JatarNg2026b} ask whether $\mathcal B_d$, the supremum of the gains over pairs with
$\deg B\le d$, is strictly increasing, and whether $\mathcal B_1=K^*$. Since $K^*=\sup_d\mathcal B_d$~\cite[Thm.~1.2]{JatarNg2026b},
$K^*=4$~\cite[Thm.~1.1]{GoelK4}, and every $\mathcal B_d$ is less than $4$~\cite[Thm.~1.3]{JatarNg2026b}, we have
$\mathcal B_d<K^*$ for every $d$. So $\mathcal B_1<K^*$, and the sequence $\mathcal B_d$ is not eventually constant; whether it is
strictly increasing remains open. They prove
$1.1018<\mathcal B_1<\sqrt{7/5}$; Theorems~\ref{thm:onezeroN18} and~\ref{thm:B1upper} give $1.1363375<\mathcal B_1<1.177801$, and Theorems~\ref{thm:twozeroK2N18} and~\ref{thm:B2upper} give
$1.1559837<\mathcal B_2<100/61$. By Corollary~\ref{cor:ES} a pair attaining $\mathcal B_d$ with $d\ge1$ is never a Blaschke model. The root
transform of Section~\ref{sec:symmetric} keeps the gain but not (M3): by Lemma~\ref{lem:rt}(c), it gives a Blaschke model
with $n$ zeros exactly when the one-zero pair passes the test of Proposition~\ref{prop:criterion} with $\sigma$ replaced by $n\sigma$.
So symmetric pairs lead to the problems $\operatorname{gain}_n$ of Corollary~\ref{cor:rt}, and whether $\sup_\Psi\operatorname{gain}_2(\Psi)>\mathcal B_1$ is open.
For the pairs of Lemma~\ref{lem:kzero} whose $Q'$ has one zero on the circle, Proposition~\ref{prop:phasecount} gives a necessary
condition for $k=3$ zeros of $B$; whether such pairs with three zeros exist is open.

\item \emph{An explicit polynomial.} The proofs that $K^*>1$ (\cite{LeanMVP2026}, and~\cite[Prop.~4.2--4.3]{JatarNg2026b} through
Runge approximation and Hilbert's lemniscate theorem) give no explicit polynomial and no bound for its degree. Adamczewski and
Colbrook~\cite{AdamczewskiColbrook2026} give one: a polynomial $p$ of degree $29\,984$ with $m(p)>1.0000004$, certified by interval
arithmetic. The larger lower bounds for $K^*$, including the one proved here, also pass from pairs to polynomials through Runge
approximation and give no explicit polynomial. Find an explicit polynomial $p$ with $m(p)>1.1$, of stated degree, with a proof of this inequality.

\item \emph{Degree dependence.} $K(n)$ is nondecreasing in $n$~\cite{Crane2006} and tends to $K^*=4$~\cite{GoelK4}. How fast? The proof in~\cite{GoelK4}
is not effective: for a given $A<4$ it gives no bound on the least $n$ with $K(n)>A$.

\item \emph{Slit maps and hooks.} By Corollaries~\ref{cor:Splus} and~\ref{cor:ratext}, an extremal $W$ that is meromorphic past the
circle is a non-rational map onto a slit plane. By Corollary~\ref{cor:hook}, such a map carries pairs $(W,C_r)$ whose zeros $r\zeta_j$ lie on fixed
radii, with $r$ close to $1$, only if some slit has an interior strict local minimum of the distance to $0$, for instance a hook; and the gains
of these pairs tend to $1$. Whether a map onto the plane minus a hook, meromorphic past the circle,
carries a pair with gain above $1.1559837$ is open; by Theorem~\ref{thm:signrule}, in the notation of Section~\ref{sec:ratext}, $Z_-$ then has a point at which $u$ is not a
nondegenerate minimum.
\end{enumerate}