The forest characterization conjecture for the strong Erdős–Hajnal property

Let $H$ be a tournament, and let an ordering of its vertices determine its backward arc digraph, whose edges are the arcs directed backward relative to that ordering. A tournament has the strong Erdős–Hajnal property if there exists $lpha>0$ such that every $H$-free tournament $T$ with more than one vertex has disjoint sets $A,B\subseteq V(T)$, every vertex of $A$ adjacent to every vertex of $B$, and $\min(|A|,|B|)\geq \alpha|T|$.

Forest characterization conjecture. If a tournament $H$ has an ordering of its vertices for which the backward arc digraph is a forest, then $H$ has the strong Erdős–Hajnal property.

Chudnovsky et al. proved the converse implication: the strong Erdős–Hajnal property forces an ordering whose backward arc digraph is a forest. The stated implication remains open.