Conjectured sufficient condition via the decomposition family

Prove or refute that, for every directed graph \vec{H} whose underlying graph H satisfies \mathcal{M}(\vec{H})=\overrightarrow{\mathcal{M}(H)}, the equality D(n,\vec{H})=2^{ex(n,H)} holds for sufficiently large n.

Background

The paper introduces directed and undirected decomposition families to control how copies of a forbidden directed graph can be embedded in approximately multipartite host graphs. The condition \mathcal{M}(\vec{H})=\overrightarrow{\mathcal{M}(H)} means, in the paper’s notation, that the relevant directed decomposition family contains all orientations of the corresponding underlying graphs.

Motivated by the exact results for anti-directed fans, the authors propose this condition as a broad sufficient criterion for equality between the orientation-counting function and the ordinary extremal bound. It remains a conjectural generalization.

References

As a start to Problem \ref{classification_prob}, and in light of the results of Theorems \ref{k-fans_thm_1} and \ref{k-fans_thm_2}, it seems reasonable to conjecture the following: Let $H$ be a directed graph with underlying graph $H$ such that $(H) = \overrightarrow{(H)}$. Then for large enough $n$, $$D(n, H) = 2{ex(n, H)}.$$

— 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 Problem \ref{classification_prob}