Conjecture on polygonal extremals in the convex $(\lambda_1,\mu_1)$-diagram

Let $\mathcal{E}^C$ be the $(\lambda_1,\mu_1)$-diagram for planar convex domains, with domains normalized to have area $1$. Let $B$ be a ball, let $T_e$ be the equilateral triangle of unit area, and write $\lambda_1(\Omega)$ and $\mu_1(\Omega)$ for the first Dirichlet and Neumann eigenvalues. For a fixed $\ell$, consider maximizers of $\mu_1(\Omega)$ among convex domains satisfying $\lambda_1(\Omega)=\ell$ and $|\Omega|=1$.

Conjecture on polygonal extremals. Except for the ball, domains on the upper boundary of $\mathcal{E}^C$ are polygonal; regular polygons lie on that upper boundary; and there exists $x_0>\lambda_1(T_e)$ such that: if $\ell\in[\lambda_1(T_e),x_0)$, a maximizer is a superequilateral triangle; if $\ell\in(x_0,+\infty)$, a maximizer is a rectangle; and at $\ell=x_0$, both a rectangle and a superequilateral triangle solve the maximization problem.

These claims are based on numerical experiments with random convex polygons. They propose a detailed description of the upper boundary and its extremal domains, but no proof or resolution is supplied in the text.