The quantum dot Dirac eigenvalue minimization conjecture

Assume that $m\geq0$. For a bounded domain $\Omega\subset\mathbb{R}^2$ with $C^2$ boundary, let $\lambda_\Omega(\theta)$ denote the first nonnegative eigenvalue of the quantum dot Dirac operator, and let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. Quantum dot Dirac eigenvalue minimization conjecture. If $\Omega$ is not a disk, then

$$\lambda_\Omega(\theta)>\lambda_D(\theta)$$

for all $\theta\in(-\frac \pi 2,\frac \pi 2)$. This is the two-dimensional counterpart of the generalized MIT bag model shape-optimization conjecture and is stated as a hot open problem in spectral geometry.