Conjectured exponential-square-root bound for the cop-pebbling number

Let $G$ be a graph on $n$ vertices, and let $\pi^{\sf c}(G)$ denote its cop-pebbling number. Cop-pebbling bound conjecture. Every graph $G$ on $n$ vertices satisfies

$$\pi^{\sf c}(G)=O(\sqrt{n}2^{\sqrt{n}}).$$

This conjecture proposes a general upper bound for the cop-pebbling number, motivated by the preceding discussion of Meyniel's conjecture; the source does not provide a resolution.