Minimal-order conjecture for graphs with large attacking-cop-number gap

Let $H$ be a graph of order $n$, and let $\operatorname{c}(H)$ and $\operatorname{cc}(H)$ denote its cop number and attacking cop number, respectively.

Minimal-order conjecture for the attacking-cop-number gap. If

$$\operatorname{cc}(H)-\operatorname{c}(H)\geq k,$$

then, for $k=2$, $n\geq 15$, while for $k=3$, $n\geq 58$.

This conjecture asserts that the known constructions with attacking cop number exceeding the cop number by two or three have minimal possible order. The source presents it as a conjecture about the minimality of the known examples.