Axisymmetry conjecture for magnetic ground-state minimizers

Let $B\geq 0$ be the strength of a constant magnetic field, and let $\Omega^*(B)$ be a domain minimizing the first eigenvalue among the admissible domains of fixed volume. The magnetic field points along the $z$-axis. Axisymmetry conjecture. For any $B\geq 0$, the minimizer $\Omega^*(B)$ is axisymmetric with respect to the $z$-axis. This is suggested by numerical optimization, including computations without an imposed symmetry assumption; whether all minimizers have this symmetry remains open.