The \overline\partial-Robin Laplacian eigenvalue minimization conjecture

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $C^2$ boundary, and for $a>0$ let $\mathcal{R}_a$ be the $\overline\partial$-Robin Laplacian on $L^2(\Omega)$, with first eigenvalue $\mu_\Omega(a)$. Let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. \overline\partial-Robin Laplacian eigenvalue minimization conjecture. If $\Omega$ is not a disk, then

$$\mu_\Omega(a)>\mu_D(a)$$

for all $a>0$. This problem is motivated by the relation between the quantum dot Dirac eigenvalues and the $\overline\partial$-Robin Laplacian eigenvalues; it is presented as an open problem based on the preceding Dirac conjecture.