Choudum–Karthick–Shalu conjecture on quadratic coloring of $P_5$-free graphs

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\omega(G)$ its clique number. A graph is $P_5$-free if it has no induced subgraph isomorphic to the five-vertex path $P_5$. Choudum–Karthick–Shalu conjecture. There exists a constant $c$ such that for every $P_5$-free graph $G$,

$$\chi(G)\le c\omega^2(G).$$

The conjecture proposes a quadratic binding function for the class of $P_5$-free graphs and is stated in the source as still open.