Non-amenability conjecture for traces of branching random walks

Let a transient branching random walk (BRW) be given on a unimodular random graph (URG), and let $\mu$ be the probability measure governing its trace, viewed as a random rooted graph. A measure on rooted graphs is non-amenable in the unimodular sense when it has positive expansion.

Non-amenability conjecture. Let $\mu$ be the measure of the trace of a transient BRW on a URG. Then $\mu$ is non-amenable.

This is known for the trace of a transient BRW on a unimodular random tree with infinitely many ends. The conjecture would imply further percolation consequences under a first-moment condition, but remains open in general.