The heroic-set conjecture for oriented forests

Let $H$ be a hero and let $F$ be an oriented forest. An oriented star is an orientation of a star, and a disjoint union of oriented stars is an oriented forest whose connected components are oriented stars. For digraphs $D_1$ and $D_2$, write $D_1+D_2$ for their disjoint union.

Heroic-set conjecture. The set

$$\{\overleftrightarrow K_2,H,F\}$$

is heroic if and only if either $F$ is the disjoint union of oriented stars or $H$ is a transitive tournament.

This conjecture proposes a characterization for the unresolved case (P2) concerning heroic sets of three digraphs. The paper completely settles the corresponding case involving a complete symmetric digraph and an independent set, while the characterization above addresses sets containing $\overleftrightarrow K_2$, a hero, and an oriented forest.