Ashbaugh–Benguria-type ratio conjecture for Dirac eigenvalues

Let $\Omega\subset\mathbb{R}^2$ be open and bounded, and let $\lambda_1(\Omega)$ and $\lambda_2(\Omega)$ denote the first two eigenvalues of the Dirac operator with infinite mass boundary conditions on $\Omega$.

Ashbaugh–Benguria-type conjecture. The disk is a solution of the maximisation problem

$$\max\left\{\frac{\lambda_2(\Omega)}{\lambda_1(\Omega)}:\ \Omega\subset\mathbb{R}^2\ \text{open and bounded}\right\}.$$

The claim is motivated by the Ashbaugh–Benguria result for the Dirichlet Laplacian. The paper reports numerical maxima close to the disk values for the tested masses $m=1$ and $m=5$, but no proof is provided.