The cop-number conjecture for connected $P_t$-free graphs

Let $G$) be a connected graph with no induced path $P_t$, where $t\geq 5$. In the cops-and-robber game on $G$, the cops and robber move along edges, and the cops capture the robber by occupying his vertex.

Cop-number conjecture. The robber can be captured by $t-3$ cops.

The conjecture asks whether the theorem proved in the paper, that $t-2$ cops capture the robber in at most $t-1$ moves, can be strengthened by one fewer cop for connected $P_t$-free graphs. Its status is not resolved in the supplied source.