Sivaraman's cop-number conjecture for path-free graphs

Let $P_k$ denote the path on $k$ vertices, let $G$ be a graph, and let $c(G)$ denote its cop number. A graph is $P_k$-free if it contains no induced subgraph isomorphic to $P_k$. Sivaraman's conjecture. For all $k\geq 5$, if $G$ is $P_k$-free, then

$$c(G)\leq k-3.$$

The conjecture proposes an improvement over the known bound that every connected $P_k$-free graph is $(k-2)$-cop win. Its general $P_5$ case was recently proved by Chudnovsky, Norin, Seymour, and Turcotte; the statement for general $k$ is therefore not uniformly open in the same form.