Diagonalizability conjecture for vertex-primitive arc-transitive digraphs

A digraph is a directed graph, and it is vertex-primitive if its automorphism group acts primitively on its vertices; it is arc-transitive if its automorphism group acts transitively on its directed edges (arcs). A digraph is diagonalizable when its adjacency matrix is diagonalizable.

Diagonalizability conjecture. Every vertex-primitive arc-transitive digraph is diagonalizable.

The conjecture is motivated by computational evidence: the paper reports no non-diagonalizable vertex-primitive arc-transitive digraph with at most 1000 vertices, while some non-diagonalizable vertex-primitive digraphs are known that are neither Cayley nor arc-transitive. A computer-free proof of the non-diagonalizability of those examples is also not known.