Polynomial χ-bounds for forest-free graphs

Let $H$ be a forest, and let $G$ be an $H$-free graph, meaning that $G$ has no induced subgraph isomorphic to $H$. Write $\chi(G)$ for its chromatic number and $\omega(G)$ for its clique number. Polynomial χ-bound conjecture for forest-free graphs. There exists a constant $c>0$, depending on $H$, such that

$$\chi(G)\leq \omega(G)^c$$

for every $H$-free graph $G$. This conjecture is presented as a consequence of combining the Gyárfás–Sumner conjecture with Esperet's conjecture; the supplied text does not state whether it has been resolved.