The locally dense graph homomorphism conjecture

For a graph $H$, call it $(\varepsilon,d)$-dense if every subset $X\subseteq V(H)$ with $|X|\geq\varepsilon|V(H)|$ spans at least $\frac d2|X|^2$ edges. For a graph $G$, write $\mathrm{e}(G)=|E(G)|$ and $\mathrm{v}(G)=|V(G)|$.

Locally dense graph homomorphism conjecture. For every graph $G$ and every $\delta,d\in(0,1)$, there exists $\varepsilon=\varepsilon(\delta,d,G)$ such that every sufficiently large $(\varepsilon,d)$-dense graph $H$ admits at least

$$(d^{\mathrm{e}(G)}-\delta)\\,\mathrm{v}(H)^{\mathrm{v}(G)}$$

homomorphisms from $G$ into $H$.

This is a dense-graph counting assertion: local density should force approximately the expected number of homomorphisms at edge density $d$. Its general validity is presented as a conjecture and remains open in the source.