Optimality of $k$ equal balls for Robin eigenvalues

Fix a dimension $d\geq 2$ and an integer $k\geq 3$. Let $\Omega\subset\mathbb{R}^d$ be a sufficiently smooth domain of volume $1$, and let $\mathcal{B}_k$ denote the disjoint union of $k$ equal balls of total volume $1$. Write $\lambda_k(\Omega,\alpha)$ for the $k$th Robin eigenvalue with boundary parameter $\alpha$. Optimality of $k$ equal balls. There exists $\alpha_k^*>0$, depending only on $k$ and $d$, such that

$$\lambda_k(\mathcal{B}_k,\alpha)\leq\lambda_k(\Omega,\alpha)$$

for all $\alpha\in(0,\alpha_k^*]$ and all such domains $\Omega$. Moreover, $\mathcal{B}_k$ is not optimal for $\alpha>\alpha_k^*$, and $\alpha_k^*\to\infty$ as $k\to\infty$. This conjecture concerns the extremal domain for Robin eigenvalues among fixed-volume domains and is presented as an open problem motivated by prior results.