Ball-to-shell transition conjecture for the first Robin eigenvalue

Let $\Omega\subset\mathbb{R}^d$ be a domain of fixed volume, and let $\lambda_1^\alpha(\Omega)$ be the first eigenvalue of the Robin problem

$$\left\{ \begin{aligned} -\Delta u &= \lambda u &&\text{in }\Omega,\\ \frac{\partial u}{\partial\nu}+\alpha u&=0 &&\text{on }\partial\Omega. \end{aligned} \right.$$

where $\alpha<0$. Ball-to-shell transition conjecture. There is a negative threshold $\alpha^*$ such that the ball maximises $\lambda_1^\alpha$ among equal-volume domains for $\alpha\in(\alpha^*,0)$; for $\alpha<\alpha^*$, the maximiser is a spherical shell whose radii increase as $\alpha$ decreases. The threshold and shell radii depend only on the dimension and the prescribed volume. Numerical simulations support this transition, but the source gives no proof.