Baur's magnetic Faber–Krahn conjecture for higher eigenvalues

Let $\Omega \subset \mathbb R^2$ be a bounded simply-connected planar domain with area $|\Omega|=1$, and let $n\in\mathbb N$. For a homogeneous magnetic field of strength $B\geq 0$, let $\lambda_n(\Omega,B)$ denote the $n$th eigenvalue of the shifted magnetic Laplacian with Dirichlet boundary conditions. Let $\mathbb D$ be the disk of unit area. Baur's conjecture. For all $B\geq 2\pi n$,

$$\lambda_n(\Omega,B)\geq\lambda_n(\mathbb D,B).$$

This conjecture proposes that, above the threshold $2\pi n$, the unit-area disk minimizes the $n$th eigenvalue among bounded simply-connected planar domains. It is motivated by numerical evidence and by the expected tendency of optimal domains for magnetic eigenvalues to become more symmetric as the magnetic field grows; its resolution is not given here.