The centered single-well and convex potential spectral-gap lower-bound conjecture

Let $V$ be a single-well potential with centered transition point $\tau=0$ or a convex potential, and let $\Lambda(V,\alpha)$ denote the fundamental spectral gap of the one-dimensional Robin Schrödinger operator with potential $V$ and Robin parameter $\alpha$. Spectral-gap lower-bound conjecture.

$$\Lambda(V,\alpha)\geq\Lambda(0,\alpha),\qquad \text{for each }\alpha\in(-\infty,\infty],$$

with equality if and only if $V$ is constant. This would extend the available lower bounds to all Robin parameters in the stated class; the source presents it as an expected conjecture, and no resolution is supplied.