Support of the limiting boundary measure of branching random walk

Candellero and Hutchcroft proved that for a supercritical branching random walk on a transient irreducible Markov chain, with an offspring law satisfying the $L\log L$ condition, the normalised occupation measures converge almost surely to a random measure $W$ on the Martin boundary, and asked for conditions under which the support of $W$ coincides almost surely with the set of accumulation points of the walk. We show that this holds whenever the boundary has a countable basis of open sets $V$ such that the exit probability $\mathbb P_x(X_\infty\in V)$ stays bounded away from zero as $x$ tends to any point of $V$. The condition holds whenever the Martin compactification is Dirichlet regular. It therefore covers simple random walk on regular trees, for which we give a self-contained proof, and finitely supported symmetric random walks on non-elementary hyperbolic groups, for every supercritical $L\log L$ offspring law. The proof needs only the integrability of the martingale limit, with no second moment, and it applies on both sides of the local survival threshold.