The convex-domain local-minimality conjecture for symmetric Maxwell eigenvalue functions

Let $B$ denote the ball in $\mathbb{R}^3$, and let the first three Maxwell eigenvalues of a domain $\Omega$ be $\lambda_1(\Omega),\lambda_2(\Omega),\lambda_3(\Omega)$. For $j=1,2,3$, let $e_j$ denote the $j$th elementary symmetric function of these eigenvalues. A domain is subject to either a fixed-volume constraint or a fixed-perimeter constraint. The convex-domain local-minimality conjecture. The ball is a local minimiser, under either volume or perimeter constraint, for the symmetric functions $e_1$, $e_2$, and $e_3$ of the first three eigenvalues among convex domains. This conjecture asks whether the local minimality suggested by the cuboid model persists in the broader class of convex domains; the corresponding claim is false without a convexity or comparable topological restriction, as nearby domains homeomorphic to a ball can have arbitrarily small higher eigenvalues.