Monotonicity of the Robin spectral ratio

Let $\Omega$ be a bounded Lipschitz domain, and let $\lambda_1(\Omega;\alpha)<\lambda_2(\Omega;\alpha)\leq\cdots$ be the Robin eigenvalues for parameter $\alpha\in\mathbb R$.

Spectral-ratio monotonicity conjecture. The map

$$\alpha\mapsto\frac{\lambda_2(\Omega;\alpha)}{\lambda_1(\Omega;\alpha)}$$

is decreasing for $\alpha>0$. The conjecture is open even for rectangles; the source notes numerical evidence that it can fail for certain acute isosceles triangles when $\alpha<0$.