The square minimisation conjecture for Dirac rectangles

Let $\Omega_{a,b}$ be a rectangle with side lengths $a,b$, and write

$$\lambda_1(a,b):=\lambda_1(\Omega_{a,b})$$

for the lowest positive eigenvalue of the Dirac operator with infinite-mass boundary conditions. Square minimisation conjecture. For every $m\geq 0$,

$$\lambda_1(a,a^{-1})\geq\lambda_1(1,1)\qquad\text{for every }a>0,$$

under the area constraint, and

$$\lambda_1(a,2-a)\geq\lambda_1(1,1)\qquad\text{for every }a\in(0,2),$$

under the perimeter constraint. This is the rectangle version of the disk optimisation conjecture, and the paper presents it as an equally challenging open problem.