Asymptotic conjecture for the bottom of the spectrum with hypersurface magnetic wells

Let $H^h$ be the operator acting in $L^2(M)$, let $\lambda_0(H^h)$ denote the bottom of its spectrum, let $k$ be the order of vanishing of the magnetic field under the current assumptions, and let $\omega_{\mathrm{min}}(B)$ be the minimum magnetic-field strength appearing there. Then the spectral-bottom asymptotic conjecture.

$$\lim_{h\to 0} h^{-\frac{2k+2}{k+2}}\lambda_0(H^h)=\hat{\nu}\, \omega_{\mathrm{min}}(B)^{\frac{2}{k+2}}.$$

This conjecture refines the approximate-eigenvalue construction for $H^{h,0}$ and predicts the precise leading-order behavior of the lowest eigenvalue as $h$ tends to zero. Its status is not resolved by the supplied source context.