Sivaraman–Testa conjecture on the cop number of induced-path-free graphs

Let $G$ be a connected graph, and let $P_t$ denote the path on $t$ vertices. A graph is $P_t$-free if it has no induced subgraph isomorphic to $P_t$, and $c(G)$ denotes its cop number. Sivaraman–Testa conjecture. For $t\geq 5$, if $G$ is $P_t$-free, then

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

This improves by one the previously known bound $c(G)\leq t-2$ for connected $P_t$-free graphs. The conjecture is presented as an improvement of that bound; its resolution is not specified in the supplied text.