Converse of k-regular Seymour-tight orientations

Prove that if an orientation O satisfies |N^-_1(O,v)| = |N^+_1(O,v)| = |N^+_2(O,v)| = k for every vertex v, then |N^-_2(O,v)| = k for every vertex v; equivalently, show that under these degree-equality conditions the converse orientation is also Seymour-tight.

Background

The converse (arc-reversal) of a Seymour-tight orientation need not be Seymour-tight in general, but this invariance holds under strong symmetry (vertex-transitivity). The authors isolate a natural degree-equality condition and conjecture that it suffices to guarantee Seymour-tightness of the converse. They verify the claim for k = 1 and k = 2 using earlier structural lemmas.

References

This naturally leads to the following question: is the converse of a Seymour-tight orientation also Seymour-tight, under the weaker assumption that every vertex has in- and out-degree $k$? We conjecture that this is the case. Let $O$ be an orientation such that $$|\NGi[-]{1}{O}{v}|=|\NGi[+]{1}{O}{v}|=|\NGi[+]{2}{O}{v}|=k $$ for all $v \in O$. Then it also holds that $|\NGi[-]{2}{O}{v}|=k$ for all $v \in O$.

— Seymour-tight orientations  (2603.29626 - Guo et al., 31 Mar 2026) in Discussion (Concluding section)