Converse-invariance conjecture for regular Seymour-tight orientations

Let $O$ be an orientation, and write $N^-_i(O,v)$ and $N^+_i(O,v)$ for the in- and out-neighbourhoods of distance $i$ from $v$. Call $O$ Seymour-tight when $|N^+_2(O,v)|=|N^+_1(O,v)|$ for every vertex $v$.

Converse-invariance conjecture. If

$$|N^-_1(O,v)|=|N^+_1(O,v)|=|N^+_2(O,v)|=k$$

for all $v e V(O)$, then

$$|N^-_2(O,v)|=k$$

for all $v\in V(O)$.

This would imply that the converse of every such regular Seymour-tight orientation is again Seymour-tight. Converse-invariance is known for vertex-transitive Seymour-tight orientations, but fails for general strongly connected Seymour-tight orientations; the stated weaker regularity condition remains open.