Aboulker et al.'s conjecture for out-stars and the directed triangle

Let $\accentset{\leftrightarrow}{K}_2$ be the complete digraph on two vertices, let $S_2^+$ be the orientation of a two-leaf star with all arcs directed outwards, and let $\vec{C}_3$ be the directed cycle on three vertices. For a family of forbidden induced digraphs, write $\operatorname{Forb}_{\operatorname{ind}}(\cdot)$ for the corresponding class and $\vec{\chi}$ for dichromatic number. Aboulker et al.'s conjecture. The two forbidden-subdigraph classes satisfy

$$\vec{\chi}(\operatorname{Forb}_{\operatorname{ind}}(\accentset{\leftrightarrow}{K}_2,S_2^+,\vec{C}_3))=\vec{\chi}(\operatorname{Forb}_{\operatorname{ind}}(\accentset{\leftrightarrow}{K}_2,S_2^-,\vec{C}_3))=2.$$

The source states that this was open for the two orientations of the three-vertex star, while the paper's conclusion says that it resolves this conjecture; hence its status is recorded as solved.