Harutyunyan–McDiarmid conjecture on acyclic sets in H-free oriented graphs

Let $H$ be an oriented graph, and let an $H$-free oriented graph be one that does not contain $H$ as a not necessarily induced subdigraph. For an oriented graph $D$, write $\vec\alpha(D)$ for its maximum acyclic set size and $\vec\chi(D)$ for its dichromatic number. Harutyunyan–McDiarmid conjecture. For every oriented graph $H$, there is $\epsilon>0$ such that every $H$-free oriented graph $D$ of order $n$ satisfies

$$\vec\alpha(D)\geq n^\epsilon\qquad\text{and}\qquad \vec\chi(D)\leq n^{1-\epsilon}.$$

This conjecture is open even when $H$ is the directed cycle of length $3$, and it strengthens the Alon–Pachs–Solymosi conjecture below.