General-domain square-root asymptotic conjecture for magnetic Dirichlet-to-Neumann ground states

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, and let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$. Let $\Lambda_{bA_0}$ be the associated magnetic Dirichlet-to-Neumann map, and let $\lambda^{\rm DN}(bA_0,\Omega)$ denote its ground-state energy. General-domain square-root asymptotic conjecture. As $b\to+\infty$,

$$\lim_{b\to+\infty}b^{-1/2}\lambda^{\rm DN}(bA_0,\Omega)=\alpha.$$

The conjecture seeks to extend the disk asymptotic to regular planar domains. It is motivated by the corresponding strong-field analysis for the magnetic Neumann Laplacian and remains presented as a hoped-for result in the source.