The rectangle lower-bound conjecture for the Cauchy-process spectral gap

Let $D\subset\mathbb{R}^2$ be a bounded convex domain symmetric with respect to both coordinate axes. Let $R$ be the smallest rectangle containing $D$ whose sides are parallel to the coordinate axes. For $0<\alpha<2$, let $\lambda_1$ and $\lambda_2$ denote the first two eigenvalues of the symmetric $\alpha$-stable process killed upon exiting the indicated domain. Rectangle lower-bound conjecture.

$$\lambda_2(R)-\lambda_1(R)\leq\lambda_2(D)-\lambda_1(D).$$

This is proposed as a sharp lower bound for the spectral gap in the symmetric stable-process setting. The source gives no resolution for $0<\alpha<2$; it contrasts this lower-bound question with the Brownian-motion case and discusses related stable-process inequalities.