Smits's conjecture on Robin spectral-gap monotonicity for convex domains

Let $\Omega$ be a bounded Lipschitz domain, let $\lambda_j^{\Omega}(\alpha)$, for $j=1,2,3,\dots$, be the Robin eigenvalues of $-\Delta$ with parameter $\alpha$, and define the fundamental spectral gap by

$$\Lambda^{\Omega}(\alpha)=\lambda_2^{\Omega}(\alpha)-\lambda_1^{\Omega}(\alpha).$$

Smits's conjecture. If $\Omega$ is a convex bounded domain, then $\Lambda^{\Omega}(\alpha)$ is an increasing function for $\alpha\geq0$. The conjecture concerns monotonicity of the fundamental Robin spectral gap in higher dimensions; the source attributes it to Smits and gives no resolution.