Asymptotic cylinder-shape conjecture for magnetic ground-state minimizers

Let $B\geq 0$ be the strength of a constant magnetic field. Let $\Omega^*(B)$ be a minimizing domain, and let $h(\Omega^*(B))$ and $R(\Omega^*(B))$ denote its height and radius. Let $h^*(B)$ and $R^*(B)$ be the height and radius of the optimal comparison cylinder. Asymptotic cylinder-shape conjecture.

$$h(\Omega^*(B))=h^*(B)(1+o(1))$$

and

$$R(\Omega^*(B))=R^*(B)(1+o(1))$$

as $B\to\infty$. Numerical minimizers increasingly resemble long cylinders with round caps, but these asymptotics remain unproved and are posed as an open problem.