Rayleigh–Bossel conjecture for the first Robin eigenvalue

Let $\Omega$ be a convex bounded planar domain, with area $A(\Omega)$, perimeter $L(\Omega)$, and first Robin eigenvalue $\lambda_1(\Omega;\alpha)$.

Rayleigh–Bossel conjecture. For each $\alpha\in\mathbb R$, the scale-invariant quantity

$$\lambda_1\bigl(\Omega;\alpha/L(\Omega)\bigr)A(\Omega)$$

is minimized when $\Omega$ is a disk. The conjecture strengthens the known Robin Faber–Krahn result by imposing convexity and scaling the Robin parameter by perimeter. The source proves the corresponding rectangle result but does not establish the assertion for arbitrary convex planar domains.