The matching-orientation conjecture for directed graphs

Let $H$ be a graph and let $\vec{H}$ be a directed graph with underlying graph $H$. For a graph or directed graph, let $\mathcal{M}$ denote the family of maximal matchings, and let $\overrightarrow{\mathcal{M}(H)}$ be the corresponding family of directed matchings. The matching-orientation conjecture. If

$$\mathcal{M}(\vec{H})=\overrightarrow{\mathcal{M}(H)},$$

then, for large enough $n$,

$$D(n,\vec{H})=2^{\mathrm{ex}(n,H)}.$$

This conjecture seeks a general criterion ensuring that the orientation-counting extremal function agrees with the ordinary extremal-number bound; the paper presents it as a difficult conjectural extension of its exact results for particular orientations, and its resolution remains open.