Asymptotic Erdős–Hajnal conjecture for forbidden order-size pairs

Let $G$ be an $n$-vertex $r$-graph, let $s(G;m)$ denote the number of values of $f$ for which $G$ contains $m$ vertices spanning exactly $f$ edges, and let $g_r(m)$ be the threshold function used in the source. A homogeneous set is a clique or an independent set. Asymptotic Erdős–Hajnal conjecture. Every $n$-vertex $r$-graph with no homogeneous sets of size $n^{\Omega(1)}$ satisfies

$$s(G;m)\geq (1-o_m(1))g_r(m).$$

The preceding non-asymptotic conjecture is false, as a construction of J. Fox gives a counterexample for $m=2r$; this asymptotic formulation is proposed as a plausible replacement. Its general status is open.