Length-scaled Robin spectral-ratio conjecture for convex planar domains

Let $\Omega$ be a convex bounded planar domain with perimeter $L=L(\Omega)$, and let $\lambda_1(\Omega;\alpha)$ and $\lambda_2(\Omega;\alpha)$ be the first two Robin eigenvalues.

Length-scaled spectral-ratio conjecture. For every $\alpha\geq-2\pi$, the quantity

$$\frac{\lambda_2(\Omega;\alpha/L)}{|\lambda_1(\Omega;\alpha/L)|}$$

is maximized when $\Omega$ is a disk. The source proves the analogous assertion for simply connected planar domains over $[-2\pi,2\pi]$ for the unratioed second eigenvalue and proves the rectangle case, but leaves this convex-domain spectral-ratio statement as a conjecture.