Full-support conjecture for semiclassical measures on complex hyperbolic quotients

Let $M$ be a compact complex hyperbolic quotient, and let $\mu$ be a semiclassical measure, meaning the semiclassical limit of a sequence of Laplacian eigenfunctions on $M$. Its support is a subset of the unit cosphere bundle $S^*M$. Full-support conjecture. For every such $\mu$,

$$\operatorname{supp}\mu=S^*M.$$

This is stronger than the corresponding statement for the pushforward measures on $M$. The preceding theorem shows that every semiclassical measure contains $S^*\Sigma$ for some compact immersed totally geodesic complex submanifold $\Sigma\subset M$; the conjecture asserts that no proper such obstruction occurs. The supplied text gives no resolution status for this conjecture.