Isosceles right-triangle conjecture for the lowest positive Dirac eigenvalue

Let $\Omega_{a,b}\subset\mathbb{R}^2$ be the right triangle with vertices $O=(0,0)$, $A=(a,0)$ and $B=(0,b)$, where $a,b>0$, and write $\lambda_1(a,b)=\lambda_1(\Omega_{a,b})$. The area is $ab/2$ and the perimeter is $a+b+\sqrt{a^2+b^2}$. For $m\geq 0$, isosceles right-triangle conjecture.

(i) If $a,b,k>0$ and $ab=k^2$, then

$$\lambda_1(a,b)\geq\lambda_1(k,k).$$

(ii) If $a\in(0,(2+\sqrt{2})k)$, $b,k>0$, and

$$a+b+\sqrt{a^2+b^2}=(2+\sqrt{2})k,$$

then

$$\lambda_1(a,b)\geq\lambda_1(k,k).$$

These assertions say that the isosceles right triangle minimises the lowest positive eigenvalue among right triangles with fixed area or perimeter. The paper presents them as the triangular analogue of the unresolved rectangle problem; the source does not provide a resolution.