Perimeter-scaled Robin eigenvalue lower bound for the disk

Let $\Omega$ be a convex bounded planar domain, let $L(\Omega)$ denote its perimeter, and let $\lambda_1(\Omega;\alpha/L(\Omega))$ be its first Robin eigenvalue with perimeter-scaled parameter. For domains of a fixed area and $\alpha\in\mathbb R$, the perimeter-scaled Robin lower-bound conjecture. The disk minimises

$$\lambda_1\left(\Omega;\frac{\alpha}{L(\Omega)}\right)$$

among all such domains. This asks for a perimeter-scaled analogue of Bossel's Rayleigh-type lower bound; its resolution is not supplied here.