The unbounded attacking-cop-number gap conjecture

For a graph $G$, let $\operatorname{c}(G)$ denote its cop number and let $\operatorname{cc}(G)$ denote its attacking cop number. Let $k$ range over the non-negative integers.

Unbounded attacking-cop-number gap conjecture. For every non-negative integer $k$, there exists a graph $H$ such that

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

This conjecture asserts that the gap between attacking cop number and cop number is unbounded. The source gives no resolution; it notes that constructing examples with gap at least $2$ has been nontrivial.