Esperet's conjecture on polynomial χ-bounds

Let $\mathcal{G}$ be a hereditary graph class, and suppose it is $\chi$-bounded: there is a function $g$ such that $\chi(G)\leq g(\omega(G))$ for every $G\in\mathcal{G}$. Esperet's conjecture. There is a polynomial function $f$ such that

$$\chi(G)\leq f(\omega(G))$$

for every $G\in\mathcal{G}$. The conjecture was disproved by Brianski, Davies, and Walczak.