Strong Erdős–Hajnal conjecture for simple digraphs

Let $T$ be a tournament, let $D$ be a simple digraph on $n$ vertices with no subdigraph isomorphic to $T$, and let $\vec{\alpha}(D)$ denote its largest acyclic set. Strong Erdős–Hajnal conjecture. For every tournament $T$ there exists $\varepsilon_T>0$ such that

$$\vec{\alpha}(D)\geq n^{\varepsilon_T}.$$

This strengthens the tournament Erdős–Hajnal conjecture from oriented tournaments to arbitrary $T$-free simple digraphs. The paper gives an upper bound $\varepsilon_T\leq 2/t$ when $T$ has $t\geq3$ vertices, while the general conjecture remains open.