Transience conjecture for the frog model on supercritical Galton–Watson trees

Let $\mathsf{GW}$ be a Galton–Watson measure defined by $(p_i)_{i\geq 0}$ with mean offspring

$$\sum_i i p_i>1.$$

For a tree $\mathsf{T}$ sampled from $\mathsf{GW}$, let $\mathrm{FM}(\mathsf{T},\eta,\mathsf{SRW})$ denote the frog model with sleeping-frog mean $\bar{\eta}$.

Transience conjecture. There exists a constant $c=c(\mathsf{GW})>0$ such that $\mathrm{FM}(\mathsf{T},\eta,\mathsf{SRW})$ is transient for $\mathsf{GW}$-almost all infinite realizations $\mathsf{T}$ whenever $\bar{\eta}<c$.

The conjecture would extend the paper’s transient-phase result from offspring distributions with bounded support and sufficiently large minimum positive offspring to every supercritical Galton–Watson measure. The source gives no resolution, so the conjecture remains open.