Alon–Pachs–Solymosi conjecture on acyclic sets in H-free tournaments

Let $H$ be a tournament. An $H$-free tournament is one that does not contain $H$ as a not necessarily induced subdigraph. For a tournament $T$, write $\vec\alpha(T)$ for its maximum acyclic set size. Alon–Pachs–Solymosi conjecture. For every tournament $H$, there exists $\epsilon>0$ such that every $H$-free tournament $T$ of order $n$ satisfies

$$\vec\alpha(T)\geq n^\epsilon.$$

This conjecture is equivalent to the Erdős–Hajnal conjecture. It is known for a few types of tournaments $H$, but remains wide open in general.