Quantum unique ergodicity conjecture for billiards

Let $\phi_n$ be Laplacian eigenfunctions on an ergodic billiard, and let a quantum limit mean a measure to which $|\phi_n|^2$ converges weakly. Quantum unique ergodicity conjecture. There is no excluded subsequence in the quantum ergodicity theorem. Equivalently, every eigenfunction sequence becomes equidistributed and there is only one quantum limit. The conjecture is proved for some arithmetic manifolds, while evidence for bouncing-ball modes in stadium billiards suggests possible failures and the general billiard case remains open.