Classification conjecture for Seymour Cayley orientations

Let $G$ be a group and let a Seymour Cayley orientation mean a Cayley orientation in which every vertex has equally large first and second out-neighbourhoods.

Cayley classification conjecture. Every Seymour Cayley orientation can be constructed by taking, possibly repeatedly, lexicographic products of empty graphs, powers of directed cycles, and regular tournaments.

The paper proves this classification for Cayley orientations of abelian groups. The conjecture asks whether the same description holds for Seymour Cayley orientations over arbitrary groups.