The trap bound for restricted cop number in diameter-two graphs

Let $G$ be a diameter-$2$ graph, let $H\subseteq G$ be a subgraph, and let $c_G(H)$ denote the least number of ordinary cops needed to catch a robber restricted to $H$, while the cops may move along all edges of $G$. Call $H$ an $s$-trap when the robber can be caught under the corresponding trap condition with parameter $s$.

Restricted cop-number trap conjecture. If $H$ is an $s$-trap in $G$, then

$$c_G(H)\leq s.$$

The paper says this conjecture could be used to improve the upper bounds, but gives no proof or counterexample.