Every geodesic spanning tree of a Cayley graph has branching number equal to the growth rate

A spanning tree T of a graph G, rooted at o, is geodesic if every vertex is at the same distance from o in T as in G. Every geodesic spanning tree of an infinite, connected, locally finite, quasi-transitive graph has branching number equal to the growth rate of the graph. For Cayley graphs this answers Question 3.12 of Lyons and Peres, Probability on Trees and Networks, where it was known for lexicographically constructed trees (Lyons and Peres; Liu and Xiang, arXiv:2010.01530). It follows that Bernoulli percolation on any such tree has critical probability one over the growth rate.