The convex-domain local-maximality conjecture for the first Maxwell eigenvalue

Let $B$ denote the ball in $\mathbb{R}^3$, and let $\lambda_1(\Omega)$ be the first Maxwell eigenvalue of a domain $\Omega$. Consider domains subject to either a fixed-volume constraint or a fixed-perimeter constraint, with locality understood in the domain topology used for the shape-optimisation problem. The convex-domain local-maximality conjecture. The ball is a local maximiser, under either volume or perimeter constraint, for the first eigenvalue among convex domains. The analogous assertion for all domains homeomorphic to a ball is expected to fail because the paper describes nearby diffeomorphic perturbations with larger first eigenvalue, but no rigorous construction is supplied there; the convex-domain assertion remains open.