Krahn–Szegő conjecture for the second Dirac eigenvalue

Let $H$ be the Dirac operator with infinite mass boundary conditions, and let $\lambda_2(\Omega)$ be its second eigenvalue on a planar bounded open set $\Omega$. Consider all such sets with unit area.

Krahn–Szegő conjecture. The minimiser of $\lambda_2(\Omega)$ is the union of two disjoint disks, each having area $\frac12$.

This is proposed as the relativistic analogue of the Krahn–Szegő inequality for the Dirichlet Laplacian. The numerical optimisation in the paper finds shapes resembling two disjoint equal-area disks, but does not establish the conjecture.