Equal-disks conjecture for sums of Robin eigenvalues

Fix positive numbers $A$ and $\alpha$. For a Lipschitz domain $\Omega\subset\mathbb{R}^2$ of area $A$, let $\lambda_j(\Omega,\alpha)$ denote its $j$th Robin eigenvalue. Let the candidate optimizer be the disjoint union of $k$ equal disks of total area $A$. Equal-disks conjecture. For $k$ sufficiently large,

$$\inf\left\{\sum_{j=1}^k\lambda_j(\Omega,\alpha):\Omega\subset\mathbb{R}^2\text{ Lipschitz},\ |\Omega|=A\right\}$$

is achieved by that union of $k$ equal disks. In particular, the quantity in the display is asymptotic to

$$\frac{2\pi^{1/2}\alpha}{A^{1/2}}k^{3/2}$$

as $k\to\infty$. The conjecture proposes the large-$k$ optimizer and its asymptotic growth rate for sums of Robin eigenvalues over all planar domains; the source presents it as an open conjecture.