Strong-recurrence threshold conjecture for branching random walks on bounded-degree graphs

Let $G$ be a graph with bounded degrees, let $P$ be the transition kernel of the underlying random walk, and for each induced connected subgraph $F\subset G$ with finite boundary let $P_F$ be the corresponding restricted transition kernel. Define

$$\widetilde{\rho}(P)=\inf_F\rho(P_F),$$

where the infimum is over all such induced connected subgraphs $F$. Let $m$ be the mean number of offspring of the branching random walk (BRW), and call the BRW strongly recurrent when it eventually visits every vertex almost surely.

Strong-recurrence threshold conjecture. Let $G$ be a graph with bounded degrees. Then the BRW is strongly recurrent if and only if

$$m>1/\widetilde{\rho}(P).$$

This conjecture was previously made in the cited work of Müller and collaborators. General criteria for strong recurrence remain incomplete, so the conjecture is open.