KNRS conjecture on locally dense graphs

Let $G$ be an $n$-vertex graph. Call $G$ $(\varepsilon,d)$-dense if, for every $U\subseteq V(G)$ with $|U|\geq \varepsilon n$,

$$e(G[U])\geq d\binom{|U|}{2}.$$

For a graph $H$, let $t_H(G)$ denote its homomorphism density and let $e(H)$ denote its number of edges.

KNRS conjecture. Let $H$ be a graph. For every $d,\eta\in(0,1)$, there exists $\varepsilon=\varepsilon(d,\eta,H)$ such that if $G$ is $(\varepsilon,d)$-dense, then

$$t_H(G)\geq (1-\eta)d^{e(H)}.$$

This conjecture asks whether one-sided local density guarantees the expected lower bound for homomorphism densities, including for nonbipartite graphs. It is attributed here to Kohayakawa, Nagle, Rödl, and Schacht; the paper notes that the general conjecture remains open, while a weaker regular version is known in some cases.