The balanced-tree conjecture for the maximum second Steklov eigenvalue

Let $D\geq 3$, and let $\mathcal{T}_S(\ell,D)$ be the set of all trees with $\ell$ leaves and maximum degree at most $D$. Construct $T_b^*(\ell,D)$ by recursively distributing the leaves as evenly as possible among at most $D$ children at the root and at most $D-1$ children at subsequent vertices, producing the most balanced tree of minimum height. Write $\lambda_2$ for the second Steklov eigenvalue.

Balanced-tree conjecture. For sufficiently large $\ell$, the tree $T_b^*(\ell,D)$ attains the maximum $\lambda_2$ among all trees in $\mathcal{T}_S(\ell,D)$.

This conjecture seeks the extremal trees for the second Steklov eigenvalue under fixed numbers of leaves and bounded maximum degree. Its resolution would identify the graphs attaining the upper bound in this class, but no resolution is supplied here.