Counting cographs in locally dense graphs

An $n$-vertex graph $G$ is $(ρ, d)$-dense if for all $U \subseteq V(G)$ of size at least $ρn$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, Rödl, and Schacht conjectured that, for any fixed grap