The strong attacking-cop-number conjecture

Let $G$ be a graph, and write $\operatorname{c}(G)$ for its cop number and $\operatorname{cc}(G)$ for its attacking cop number.

Strong attacking-cop-number conjecture. For all integers $k \geq 4$ there exists a graph $H$ such that $\operatorname{c}(H)=k$ and

$$\operatorname{cc}(H)=2\operatorname{c}(H).$$

The conjecture asks whether the general upper bound $\operatorname{cc}(G)\leq 2\operatorname{c}(G)$ is attained for every cop number at least four. The source states that its truth is unclear and that constructing examples with attacking cop number exceeding the cop number by at least two is nontrivial.