Reverse Bossel–Daners conjecture for negative Robin parameter

Let $\Omega\subset\mathbb R^n$ be an open bounded Lipschitz set, let $\Omega^\sharp$ be a ball with the same measure as $\Omega$, and let $\alpha<0$. Define the first Robin eigenvalue by

$$\lambda_R(\Omega)=\min_{\substack{w\in H^1(\Omega)\w\ne0}}\frac{\displaystyle\int_\Omega |Dw|^2\,dx+\alpha\displaystyle\int_{\partial\Omega}w^2\,d\mathcal H^{n-1}}{\displaystyle\int_\Omega w^2\,dx}.$$

Reverse Bossel–Daners conjecture. One has

$$\lambda_R(\Omega)\le\lambda_R(\Omega^\sharp).$$

This is the negative-parameter counterpart of the Bossel–Daners inequality, which has the opposite inequality for $\alpha\ge0$; the source states that this conjecture is still open.