The acyclic hero characterization for tournaments

Let $T$ and $H$ be tournaments. A tournament is an acyclic hero if there is a constant $c_H$ such that every $H$-free tournament has acyclic dichromatic number at most $c_H$. For tournaments $H_1,H_2$, let $H_1\Rightarrow H_2$ denote the tournament formed from disjoint copies by orienting every arc from $H_1$ to $H_2$, and let $\Delta(H_1,H_2,H_3)$ denote the cyclic composition of three tournaments. Write $k$ for the transitive tournament $TT_k$. For every $k$, define

$$H_k=TT_k\Rightarrow(\Delta(k,1,1)\Rightarrow TT_k),$$

and let $\mathcal{AH}$ be the class of tournaments that occur as subtournaments of some $H_k$.

The acyclic hero conjecture. A tournament $H$ is an acyclic hero if and only if $H\in\mathcal{AH}$.

The forward implication is proved in the paper, as are several examples of tournaments in $\mathcal{AH}$ that are acyclic heroes. The converse, and hence the characterization, remains open.