Erde–Kang–Lehner–Mohar–Schmid conjecture for cop numbers of uniform hypergraphs

Let $H$ be a $k$-uniform hypergraph, meaning that every edge of $H$ has $k$ vertices, and let $c(H)$ denote its cop number: the minimum number of cops needed for the Cop Player to have a winning strategy. Assume that $H$ is connected and has $n$ vertices.

Erde–Kang–Lehner–Mohar–Schmid conjecture.

$$c(H)=O\!\left(\sqrt{\frac{n}{k}}\right).$$

This conjecture generalizes Meyniel's conjecture from graphs to uniform hypergraphs. The source presents it as a recent conjecture and gives no resolution.