Counting cographs in locally dense graphs

An $n$-vertex graph $G$ is $(\rho ,d)$-dense if for all $U\subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}\mathrm{\mid }U{\mathrm{\mid }}^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, Rödl, and Schacht conjectured that, for any fixed graph $F$, the number of homomorphic copies of $F$ in a given $n$-vertex $(o(1),d)$-dense graph is at least $(1-o(1))d^{e(F)}{n}^{v(F)}$. Despite considerable effort, only a few graphs are known to satisfy the conjecture. We find a new broad class of graphs that satisfy the conjecture, including cographs.