Esperet's conjecture on polynomially chi-bounded hereditary classes

A graph class is hereditary if it is closed under taking induced subgraphs. It is chi-bounded if there is a function $f$ such that $\chi(G)\leq f(\omega(G))$ for every graph $G$ in the class, and it is poly-chi-bounded if such a function can be chosen to be a polynomial.

Esperet's conjecture. Every chi-bounded hereditary class is poly-chi-bounded.

The source states that this conjecture was recently disproved; the current question is to identify which hereditary classes are poly-chi-bounded.