The reverse Faber–Krahn conjecture for the magnetic Neumann Laplacian

Let $\beta>0$, and let $\Omega\subset\mathbb R^2$ be an open, bounded and simply connected set with a $C^\infty$ smooth boundary. Let $\Omega^*$ be the disk in $\mathbb R^2$ centered at the origin such that $|\Omega^*|=|\Omega|$. Reverse Faber–Krahn conjecture. For all $\beta>0$,

$$\lambda_1^N(\Omega,\beta)\le\lambda_1^N(\Omega^*,\beta),$$

with equality if and only if $\Omega$ is a disk. This conjecture is suggested by results of Fournais and Helffer showing the reverse inequality for sufficiently small and sufficiently large magnetic fields when $\Omega$ is not a disk; the assertion for every positive $\beta$ remains open.