Transience conjecture for simple random walk on branching-random-walk traces

Let a transient branching random walk (BRW) be given on a graph, and let its trace be the graph formed by the vertices and edges visited by the BRW. Let simple random walk (SRW) on this trace be the usual nearest-neighbor random walk.

Trace-transience conjecture. The trace of a transient BRW is almost surely transient for SRW.

The source proposes this as a generalization of a theorem proved there for traces in the unimodular setting. Its validity for arbitrary underlying graphs remains open.