Three-in or three-out bowtie orientations

Prove or refute that, for every orientation B of the bowtie graph F_{2,3} having exactly three edges directed toward its center vertex or exactly three edges directed away from its center vertex, the equality D(n,B)=2^{ex(n,B)} holds for sufficiently large n.

Background

The paper separates the orientations of the bowtie graph F_{2,3} into anti-directed orientations, orientations covered by Proposition \ref{not_semi_anti_directed_prop}, and a remaining class. The remaining class consists of orientations with exactly three edges entering or exactly three edges leaving the center.

The paper conjectures that these remaining orientations obey the standard extremal formula, although no proof is provided. This would resolve the last bowtie cases left untreated by the preceding results.

References

Let $B$ be an orientation of $B$ such that there are exactly three edges oriented towards the center vertex (or exactly three edges oriented away from the center vertex). Then $$D(n, B) = 2{ex(n, B)}$$

— Orientations of graphs omitting non-edge-critical directed graphs  (2502.21287 - Sheats, 28 Feb 2025) in Section 5, subsection “Questions and open problems,” Conjecture following Conjecture \ref{all_bowties_conj}