Descendant sets in infinite, primitive highly arc transitive digraphs with prime power out-valency

The descendant set desc(α) of a vertex α in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from α. We study the structure of descendant sets Γ in an infinite, primitive, highly arc transitive digraph with out-valency pk, where p is a prime and k≥1. It was already known that Γ is a tree when k=1 and we show the same holds when k=2. However, for k≥3 there are examples of infinite, primitive highly arc transitive digraphs of out-valency pk whose descendant sets are not trees, for some prime p.