The subdivided complete-graph conjecture for the cop number

Let $K_n$ be a complete graph, and let $G$ be any graph obtained from $K_n$ by subdividing its edges, possibly a non-uniform number of times. The cop number of $G$, denoted by $\operatorname{cop}(G)$, is the minimum number of cops needed to capture a robber on $G$.

Subdivided complete-graph conjecture. For every such graph $G$,

$$\operatorname{cop}(G)\in O(\sqrt n).$$

This conjecture would imply Meyniel's conjecture that every graph on $n$ vertices has cop number in $O(\sqrt n)$. The surrounding discussion establishes an upper bound of at most $2$ for the uniformly subdivided complete graph considered in the construction, but does not resolve the conjecture for arbitrary subdivisions.