Andrews–Clutterbuck–Hauer Robin gap conjecture

Fix $\alpha>0$ and dimension $n\geq2$. Let $\Omega\subset\mathbb R^n$ be a convex bounded domain of diameter $D$, and let $\lambda_1(\Omega;\alpha)$ and $\lambda_2(\Omega;\alpha)$ be its first two Robin eigenvalues.

Andrews–Clutterbuck–Hauer conjecture. The Robin spectral gap is minimized by the degenerate box, identified with the line segment $(0,D)$:

$$\lambda_2(\Omega;\alpha)-\lambda_1(\Omega;\alpha)>\lambda_2((0,D);\alpha)-\lambda_1((0,D);\alpha).$$

This extends the known Neumann and Dirichlet gap inequalities for convex domains. The general Robin assertion is presented as open, although the source proves it for rectangular boxes.