Magnetic second-eigenvalue minimizer phase-transition conjecture

Let $B\geq0$ be a constant magnetic field, let $\Omega\subset\mathbb{R}^2$ be a bounded open domain with $|\Omega|=1$, and let $\lambda_2(\Omega,B)$ be the second eigenvalue of the magnetic Dirichlet Laplacian. Magnetic second-eigenvalue phase-transition conjecture. The minimizing domain changes at the critical field $B=4\pi$: it is two disjoint disks of equal measure for $B\leq4\pi$, and a single disk for $B\geq4\pi$. Numerical evidence suggests a sudden transition at the threshold, but the conjecture remains open.