Benjamini's conjecture on ends of traces of branching random walks

Let $G$ be a non-amenable vertex-transitive graph, and let $P$ be the symmetric transition kernel of the underlying random walk. A branching random walk (BRW) on $G$ is transient when its trace is transient for the relevant random-walk structure.

Benjamini's conjecture. The trace of a transient BRW on $G$ has infinitely many ends.

This conjecture concerns the geometry of branching-random-walk traces on non-amenable graphs. The claim is known in several settings, including groups with infinitely many ends and certain subcritical or critical cases, but the stated generality remains open, including for important examples such as Cayley graphs of Gromov-hyperbolic groups.