The square-root bound for the cop number of diameter-two graphs

Let $G$ be a graph of diameter $2$ and order $n$, and let $c(G)$ denote its cop number.

Square-root bound conjecture. One should have

$$c(G)\leq \sqrt{n}.$$

The paper proves the weaker bound $c(G)\leq \sqrt{2n}$ and notes that Moore graphs have cop number $\sqrt{n-1}$, so this conjecture would be close to best possible.