The Ramsey-type bound conjecture for multipartite-free chi-bounded classes

Let $\mathcal G$ be a $\chi$-bounded graph class. For integers $m,s\ge1$, let $mK_s$ be the disjoint union of $m$ copies of $K_s$, and let $\overline{mK_s}$ be its complement. Let $R(s,w)$ denote the least integer $n\ge1$ such that every $n$-vertex graph has a stable set of size $s$ or a clique of size $w$.

Ramsey-type bound conjecture. For every $\chi$-bounded class $\mathcal G$ and every two integers $m,s\ge1$, there exists $C=C(\mathcal G,m,s)\ge1$ such that every $\overline{mK_s}$-free graph $G\in\mathcal G$ satisfies

$$\chi(G)\le C\cdot R(s,\omega(G)+1).$$

For $\chi$-dense classes, a bound of this form follows from the density framework, whereas its asserted extension to all $\chi$-bounded classes remains open.