Supersaturation conjecture for clique counts in $K_{r+1}$-free graphs

Let $r\geq s\geq 3$, let $\epsilon>0$, and let $G$ be a graph with $m=e(G)$ edges. Write $\operatorname{k}_s(G)$ for the number of copies of $K_s$ in $G$. Supersaturation conjecture. There exist $\delta>0$ and $m_0$ such that, whenever $m\geq m_0$ and

$$\operatorname{k}_s(G)\geq \operatorname{mex}_{K_s}(m,K_{r+1})+\epsilon m^{s/2},$$

then $G$ contains at least $\delta m^{(r+1)/2}$ copies of $K_{r+1}$. This predicts a quantitative supersaturation phenomenon: exceeding the extremal $K_s$ count for $K_{r+1}$-free graphs by a term of order $m^{s/2}$ forces many copies of $K_{r+1}$; the supplied text gives no resolution status.