Full-support conjecture for eigenfunction limits on negatively curved manifolds

Let $(M,g)$ be a compact connected Riemannian manifold of negative sectional curvature. A weak limit $\tilde\mu$ is a weak-* limit of the probability measures $|u_j|^2\,d\operatorname{vol}_g$ associated with a sequence of Laplacian eigenfunctions. Full-support conjecture. Each weak limit $\tilde\mu$ satisfies

$$\operatorname{supp}\tilde\mu=M.$$

Equivalently, for each nonempty open set $\Omega\subset M$ there exists a constant $c_\Omega>0$ such that

$$\|u\|_{L^2(\Omega)}\geq c_\Omega\|u\|_{L^2(M)}$$

for any Laplacian eigenfunction $u$. The conjecture was proved by Dyatlov and Jin for compact hyperbolic surfaces; the present paper proves the corresponding result for compact complex hyperbolic quotients.