Conformal-radius maximality conjecture for the second Robin eigenvalue

Suppose $F:\mathbb D\to\Omega$ is a conformal map of the unit disk onto a Jordan–Lipschitz domain $\Omega$, and let $L(\Omega)$ denote the perimeter of $\Omega$. For $\alpha>0$, the conformal-radius conjecture for the second Robin eigenvalue. The scale-invariant quantity

$$\lambda_2\bigl(\Omega;\alpha/L(\Omega)\bigr)|F'(0)|^2$$

is maximal when $F$ is linear and $\Omega$ is a disk. The conjecture extends the established conformal-radius maximality result for the first Robin eigenvalue to the second eigenvalue. Its Dirichlet limiting case is known, but the Robin statement remains open in the source.