Freitas–Laugesen conjecture for the second Robin eigenvalue on convex domains

Let $\Omega\subset\mathbb R^n$ be a convex bounded domain, with volume $V=V(\Omega)$, surface area $S=S(\Omega)$, and second Robin eigenvalue $\lambda_2(\Omega;\alpha)$. Let $\mathbb B$ be the unit ball, with volume $V(\mathbb B)$ and surface area $S(\mathbb B)$.

Freitas–Laugesen conjecture. The ball maximizes

$$\lambda_2\bigl(\Omega;\alpha V^{1-2/n}/S\bigr)V^{2/n}$$

among such domains whenever

$$-1\leq\alpha\frac{V(\mathbb B)^{1-2/n}}{S(\mathbb B)}\leq0.$$

Consequently, the ball also maximizes $\lambda_2(\Omega;\alpha/S^{1/(n-1)})V^{2/n}$ for a suitable range of $\alpha$. The source proves related bounds for simply connected planar domains and states that higher-dimensional generalization remains open.