Universality conjecture for Gaussian time-frequency localization eigenvalues

Let $\Omega\subset\mathbb{R}^2$ be compact, regular closed, and have finite boundary. Let $g_0$ be the standard Gaussian window, let $A_{R\Omega}^{g_0}$ denote the localization operator with symbol $R\Omega$, and let $\lambda_k^{R\Omega}$ be its $k$-th eigenvalue. Write $|\Omega|$ for the area of $\Omega$ and $|\partial\Omega|$ for the length of its boundary. The complementary error function is

$$\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}\,dt.$$

Universality conjecture. For every such $\Omega$,

$$\mathopen{}\mathclose\bgroup\originalleft|\lambda_k^{R\Omega}-\frac{1}{2}\operatorname{erfc}\mathopen{}\mathclose\bgroup\originalleft(\sqrt{2\pi}\frac{k-R^2|\Omega|}{R|\partial\Omega|}\aftergroup\egroup\originalright)\aftergroup\egroup\originalright|=O\mathopen{}\mathclose\bgroup\originalleft(\frac{1}{R}\aftergroup\egroup\originalright)$$

as $R\to\infty$. Equivalently, the $k$-th eigenvalue converges to

$$\frac{1}{2}\operatorname{erfc}\mathopen{}\mathclose\bgroup\originalleft(\sqrt{2\pi}\frac{k-R^2|\Omega|}{R|\partial\Omega|}\aftergroup\egroup\originalright).$$

For rotationally invariant symbols this behavior is obtained explicitly, while numerical tests support its universality for a diverse collection of sets. The conjecture remains open for general symbols with the standard Gaussian window.