E. Oudet's regular-polygon conjecture for the Steklov maximizer

Let $\mathcal{C}_{N}$ be the class of convex polygons in $\mathbb{R}^2$ with at most $N$ sides and perimeter $2\pi$, and let $\Omega^*$ be a maximizer of the first nonzero Steklov eigenvalue $\sigma_1$ over this class. The preceding existence and non-degeneracy conjecture asserts that such a maximizer exists and has exactly $N$ sides. E. Oudet's regular-polygon conjecture. The maximizer $\Omega^*$ must be a regular $N$-gon. This stronger Pólya-type conjecture was proposed by E. Oudet at an AIM workshop in 2018. The supplied context gives no resolution status.