Polynomial chi-boundedness conjecture for path-free graphs

For a graph $G$, let $\chi(G)$ denote its chromatic number and let $\omega(G)$ denote its clique number. A graph is $P$-free if it contains no induced copy of the path $P$. Path-free polynomial chi-boundedness conjecture. For every integer $k$, there is a constant $C_k>0$ such that every graph $G$ satisfying

$$\chi(G)\geq \omega(G)^{C_k}$$

contains an induced path of length $k$. Equivalently, for every fixed path, the class of graphs excluding that path as an induced subgraph is polynomially chi-bounded. The general chi-boundedness of path-free classes is known only in limited cases; the paper notes a bound for $P_5$-free graphs, while the stated polynomial form remains open.