The C4-free subclass conjecture for chi-bounded graph classes

Let $\mathcal G$ be a class of graphs. Its $C_4$-free subclass consists of the graphs in $\mathcal G$ with no induced cycle on four vertices. A graph class is linearly $\chi$-bounded if there is a constant $C\ge1$ such that every graph $G$ in the class satisfies $\chi(G)\le C\omega(G)$.

C4-free subclass conjecture. The $C_4$-free subclass of every $\chi$-bounded class is linearly $\chi$-bounded.

This would generalize known results for several $C_4$-free subclasses, while the statement remains open for arbitrary $\chi$-bounded classes.