The strong four-vertex tournament conjecture

Let $\accentset{\leftrightarrow}{K}_2$ be the complete digraph on two vertices, let $S_2^+$ be the two-out-star, and let $\vec{K}_4^s$ be the unique strong tournament on four vertices. For a family of forbidden induced digraphs, write $\operatorname{Forb}_{\operatorname{ind}}(\cdot)$ for the corresponding class and $\vec{\chi}$ for dichromatic number. Strong four-vertex tournament conjecture.

$$\vec{\chi}(\operatorname{Forb}_{\operatorname{ind}}(\accentset{\leftrightarrow}{K}_2,S_2^+,\vec{K}_4^s))=3.$$

This is presented as the smallest open case of the broader question whether $\{\accentset{\leftrightarrow}{K}_2,S_2^+,H\}$ is heroic for every hero $H$; the authors state that a new method may be required.