Berry's random wave conjecture for chaotic billiards

Let $\Omega\subset\mathbb{R}^2$ be a chaotic billiard, so its billiard flow is ergodic and has positive topological entropy. Let $\{u_n\}$ be an orthonormal sequence of eigenfunctions, and let $\sigma^\Omega(\{u_{n_j}\}_{j=1}^\infty)$ denote the accumulation-point set of the associated local measures on the space

$$\mathcal{M}:=\{\varphi\in C^\infty(\mathbb{R}^2,\mathbb{R}):\Delta\varphi+\varphi=0\}.$$

Let $\mu_{\mathrm{RMW}}$ be the Gaussian probability measure on $\mathcal{M}$ associated with the random monochromatic wave. Berry's random wave conjecture. There exists a subsequence of positive integers $\{n_j\}_{j=1}^\infty$ with full density such that

$$\sigma^\Omega(\{u_{n_j}\}_{j=1}^\infty)=\{\mu_{\mathrm{RMW}}\}.$$

This conjecture predicts the typical Planck-scale structure of high-energy eigenfunctions in chaotic systems. It is a challenging open problem in quantum chaos; no counterexamples are known, but rigorous local-limit results are essentially restricted to the flat two-torus.