Second Robin eigenvalue conjecture for planar exterior domains

Let $\Omega\subset\mathbb{R}^2$ be a simply connected bounded domain, let $\Omega^{\rm c}$ denote its exterior, and let $\alpha<0$. Define the second variational Robin level by

$$\lambda_2^\alpha(\Omega^{\rm c}):= \inf_{\substack{\mathcal L\subset H^1(\Omega^{\rm c})\\ \dim\mathcal L=2}}\ \sup_{u\in\mathcal L\setminus\{0\}} \frac{\displaystyle\int_{\Omega^{\rm c}}|\nabla u|^2+\alpha\int_{\partial\Omega}|u|^2}{\displaystyle\int_{\Omega^{\rm c}}|u|^2}.$$

Second Robin eigenvalue conjecture. If $\mathcal B$ is a disk, then

$$\lambda_2^\alpha(\Omega^{\rm c})\leq\lambda_2^\alpha(\mathcal B^{\rm c})$$

when $\mathcal B$ has the same area as $\Omega$, and also when $\mathcal B$ has the same perimeter as $\Omega$.

For negative $\alpha$ of sufficiently large magnitude, this variational level is the second discrete eigenvalue of the Robin Laplacian; for small $|\alpha|$, it is the bottom of the essential spectrum. The conjecture proposes that the exterior disk maximises the second level under both isochoric and isoperimetric constraints.