Containability bound by cop number and maximum degree

Let $G$ be a graph. Write $\xi(G)$ for its containability number, the minimum number of cops needed to contain the robber, $c(G)$ for its cop number in the original Cops and Robbers game, and $\Delta(G)$ for its maximum degree. Containability bound. For every graph $G$,

$$\xi(G)\leq \Delta(G)c(G).$$

The paper explicitly says that the authors were unable to prove or disprove this bound; it is presented as a conjecture and remains open.