Kohayakawa–Nagle–Rödl–Schacht conjecture for locally dense graphs

Let $H$ and $G$ be graphs, let $t(H,G)$ denote the homomorphism density of $H$ in $G$, and call an $n$-vertex graph $G$ $(\rho,d)$-dense when every subset $X$ of at least $\rho n$ vertices spans at least $\frac{d}{2}|X|^2$ edges. Kohayakawa–Nagle–Rödl–Schacht conjecture. For every graph $H$ and all reals $d,\varepsilon>0$, there exists a $\rho=\rho(d,H,\varepsilon)>0$ such that

$$t(H,G)\geq (1-\varepsilon)\cdot d^{|E(H)|}$$

holds for every sufficiently large $(\rho,d)$-dense graph $G$. This conjecture extends the random-graph lower bound from bipartite graphs to locally dense host graphs and is a central open problem in extremal graph theory; the source records only special classes of graphs for which it is known.