Non-disk minimiser conjecture for the third Dirac eigenvalue

Let $H$ be the Dirac operator with infinite mass boundary conditions, and let $\lambda_3(\Omega)$ denote its third eigenvalue on a planar bounded open set $\Omega$. Among all such domains with a fixed area, consider the minimisation problem.

Non-disk minimiser conjecture. There exist masses $m\geq0$ such that the minimiser of $\lambda_3(\Omega)$ is not the disk.

The conjecture contrasts with the conjectured disk minimiser for the third eigenvalue of the Dirichlet Laplacian. Numerical experiments in the paper provide support for the claim, including an optimisation at $m=1$, but no rigorous resolution is given.