Weak Meyniel's conjecture on the cop number of arbitrary graphs

Let $G$ be a graph of order $n$, and let $c(G)$ denote its cop number. Weak Meyniel's conjecture. There exists an $\varepsilon>0$ such that

$$c(G)=O(n^{1-\varepsilon}).$$

This weaker form of Meyniel's conjecture, also known as the Soft Meyniel conjecture, remains widely open and seeks a sublinear upper bound on the cop number of every graph.