Strong recurrence conjecture for branching random walks on traces

Let a transient branching random walk (BRW) be given, and let its trace be the graph formed by the vertices and edges visited by the BRW. Consider another BRW whose underlying graph is almost every such trace.

Trace strong-recurrence conjecture. Every BRW on almost every trace of a transient BRW is strongly recurrent.

The source states this in a stronger form than an earlier conjecture of Benjamini and Gurel-Gurevich. The general assertion remains open, although the paper proves strong recurrence in several unimodular settings.