Polynomial χ-boundedness conjecture for bounded merge-width classes

A graph class is χ-bounded if there is a function $f$ such that every graph $G$ in the class satisfies $\chi(G)\leq f(\omega(G))$, where $\chi(G)$ is the chromatic number and $\omega(G)$ is the clique number. It is polynomially χ-bounded if $f$ can be chosen to be a polynomial. Polynomial χ-boundedness conjecture for bounded merge-width classes. Every class of graphs with bounded merge-width is polynomially χ-bounded. Bounded merge-width classes are already known to be χ-bounded, and the conjecture asks for the stronger polynomial dependence on clique number. The stated paper proves χ-boundedness, but does not establish this polynomial strengthening.