Existence and non-degeneracy conjecture for the Steklov maximizer among convex polygons

Let

$$\mathcal{C}_{N}=\{\Omega\in\mathbb{R}^2:\Omega\text{ is a convex polygon with at most }N\text{ sides and }\operatorname{Per}(\Omega)=2\pi\},$$

where $\operatorname{Per}(\Omega)$ denotes the perimeter, and let $\sigma_1(\Omega)$ be the first nonzero Steklov eigenvalue. Define

$$\sigma^*(N):=\sup_{\Omega\in\mathcal{C}_{N}}\sigma_1(\Omega).$$

Existence and non-degeneracy conjecture. The supremum is attained by a convex polygon $\Omega^*$ with exactly $N$ sides. In particular, the maximizer cannot be a degenerate polygon with fewer than $N$ sides. This conjecture asserts existence of an extremizer in the fixed-perimeter class and rules out loss of sides through degeneration. The supplied context gives no resolution status.