Concavity conjecture for the second Robin eigenvalue

Let $\Omega$ be a convex bounded domain, and let $\lambda_2(\Omega;\alpha)$ denote its second Robin eigenvalue for parameter $\alpha\in\mathbb R$.

Second-eigenvalue concavity conjecture. The function $\lambda_2(\Omega;\alpha)$ is concave in $\alpha>0$. Concavity is known for domains with a suitable symmetry, including balls and rectangular boxes; the source also proves strict concavity for the first two eigenvalues of rectangular boxes for every real $\alpha$. Nonconcavity may occur for negative $\alpha$ on general convex domains, so the positive-parameter assertion remains open.