De Joannis de Verclos–Kang–Pastor conjecture for squares of claw-free graphs

Let $G$ be a claw-free graph, meaning that $G$ has no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. Let $G^2$ be the graph obtained from $G$ by adding an edge between every pair of vertices joined by a two-edge path, and let $\omega(G)$ and $\chi(G^2)$ denote its clique and chromatic numbers, respectively.

De Joannis de Verclos–Kang–Pastor conjecture. For any claw-free graph $G$,

$$\chi(G^2) \leq \frac14(5\omega(G)^2 -2\omega(G)+1)$$

if $\omega(G)$ is odd, and

$$\chi(G^2) \leq \frac54\omega(G)^2$$

otherwise.

This conjecture strengthens the Erdős–Nešetřil conjecture from line graphs to all claw-free graphs. The source describes it as a conjecture; the supplied status is unknown, so its resolution is not established here.