Curvature-corrected asymptotic conjecture for magnetic Dirichlet-to-Neumann ground states

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

$$b^{-1/2}\lambda^{\rm DN}(bA_0,\Omega)=\alpha-\frac{\alpha^2+2}{6}\max_{x\in\partial\Omega}\kappa_x\,b^{-1/2}+o(b^{-1/2}).$$

This refines the general-domain square-root law by predicting the first curvature-dependent correction. The source presents it as a further conjectural two-term expansion motivated by the disk computation.