Aboulker et al.'s conjecture on heroic triples of oriented star forests

An oriented graph is a digraph with no digons, and its dichromatic number $\vec{\chi}(D)$ is the least number of acyclic sets partitioning its vertex set. A digraph is a heroic set obstruction when the class of digraphs excluding every member of the set as an induced subdigraph has bounded dichromatic number; a hero is a tournament with this property when it is the sole nontrivial forbidden tournament. Aboulker et al.'s conjecture. For every orientation of a star forest $F$ and every hero $H$, the oriented graphs excluding $F$ and $H$ as induced subdigraphs have bounded dichromatic number. This is the third type in the proposed classification of heroic sets of size three; the source notes that it is known for transitive $H$ and when $F$ has at most two vertices, but remains open in general.