The cop-number conjecture for graphs without long holes

Let $G$ be a graph without a hole of length at least $t$, where $t\geq 6$. A cop is one player in the cops-and-robber game on $G$, and the cop number is the minimum number of cops needed to capture the robber.

Cop-number conjecture. The graph $G$ can be won by $t-4$ cops.

The preceding argument gives the bound $t-3$ cops, so this conjecture proposes improving it by one. Its first case, that two cops suffice for graphs without holes of length at least six, is highlighted as particularly interesting; the source states that the conjecture remains open.