Robin spectral-ratio conjecture for the ball

Fix $n\geq2$. Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain of volume equal to that of the ball $B(R)$, and let $\lambda_1(\Omega;\alpha)$ and $\lambda_2(\Omega;\alpha)$ be the first two Robin eigenvalues.

Ball spectral-ratio conjecture. For $\alpha\geq-1/R$ with $\alpha\neq0$, the ratio

$$\frac{\lambda_2(\Omega;\alpha)}{|\lambda_1(\Omega;\alpha)|}$$

is maximized when $\Omega=B(R)$. The conjecture is known at the endpoint $\alpha=-1/R$ and for sufficiently small negative parameter in suitable smooth planar settings; its limiting Neumann case also follows from classical isoperimetric and Szegő–Weinberger results. The full assertion in the stated range remains open.