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

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