Conjecture on the semiclassical ground-state limit for non-degenerate minima

Let $M$ be a connected manifold and let $V$ be a potential whose minima are non-degenerate in the sense of Shubin's Condition C. Let $\lambda_t$ denote the relevant analytic eigenvalue branch in the semiclassical parameter $t$. Semiclassical ground-state limit conjecture. Then $t^{-2}\lambda_t$ converges to the absolute minimum of $V$. The claim extends the one-dimensional conclusion suggested by the simplicity and ordering of the spectrum; its general validity under the stated non-degeneracy and connectedness assumptions is left as a conjectural assertion.