The multi-robber damage conjecture on saving three vertices

Let $G$ be a graph with maximum degree $\Delta(G)$, and let $s$ be a positive integer. In the one-cop, $s$-robber damage game, the cop saves $k$ vertices if the optimal damage satisfies $\operatorname{dmg}(G;s)\leq |V(G)|-k$.

Multi-robber damage conjecture. For all $s>2$, if

$$\Delta(G)\geq \binom{s}{2}+2,$$

then the cop can save three vertices against $s$ robbers.

For two robbers, the corresponding result is known under the weaker condition $\Delta(G)\geq 3$. The conjecture proposes a sufficient maximum-degree condition for saving three vertices with more than two robbers; its status is not established in the supplied text.