The cocktail-party-free subclass conjecture for chi-bounded graph classes

For an integer $m\ge1$, let $mK_2$ be the disjoint union of $m$ copies of $K_2$, and let $\overline{mK_2}$ be its complement. A graph class is linearly $\chi$-bounded if there is a constant $C\ge1$ such that every graph $G$ in the class satisfies $\chi(G)\le C\omega(G)$.

Cocktail-party-free subclass conjecture. For every integer $m\ge1$, the $\overline{mK_2}$-free subclass of every $\chi$-bounded class is linearly $\chi$-bounded.

This extends the preceding $C_4$-free subclass conjecture and is known for subclasses of $\chi$-dense classes, but remains open for all $\chi$-bounded classes.