Distance-transitive digraphs are Seymour-tight

Prove that every distance-transitive digraph is a Seymour-tight orientation; that is, for every vertex v the sizes of the first and second out-neighbourhoods are equal.

Background

The paper notes that known distance-transitive examples (directed cycles, Paley tournaments, and lexicographic products with empty graphs) are Seymour-tight, and refers to nonexistence results restricting other possibilities. Motivated by this, the authors conjecture that the entire class of distance-transitive digraphs must be Seymour-tight.