Classification of Seymour Cayley orientations

Determine whether every Seymour Cayley orientation can be constructed by taking (possibly repeated) lexicographic products of empty digraphs, k-th powers of directed cycles, and regular tournaments.

Background

After classifying all Seymour orientations that are Cayley digraphs of abelian groups via Kemperman’s theorem, the authors ask whether the same purely combinatorial description extends to all Cayley groups. A positive resolution would characterize Seymour Cayley orientations by a small set of building blocks closed under lexicographic product.