Linear cop-number conjecture for claw-free path-free graphs

Let $P_k$ denote the path on $k$ vertices, let $\operatorname{claw}=K_{1,3}$, and let $c(G)$ denote the cop number of a graph $G$. A graph is $(P_k,\operatorname{claw})$-free if it contains neither an induced $P_k$ nor an induced claw. Linear cop-number conjecture. There exists an $\varepsilon>0$ such that for every integer $k\geq 1$, there is a $(P_k,\operatorname{claw})$-free graph $G$ with

$$c(G)\geq \varepsilon k.$$

This conjecture asks for a linear lower bound on the cop number even after forbidding both a path and a claw. The source notes that existing random $P_k$-free examples with large cop number are typically not claw-free, and presents this as an easier-to-prove alternative to determining the precise asymptotic bound; it remains open.