Mubayi's supersaturation conjecture for stable non-r-partite hypergraphs

Fix $r\ge 2$. An $r$-uniform hypergraph, or $r$-graph, is a family of $r$-subsets of a vertex set; write $|\mathcal H|$ for its number of edges. For an $r$-graph $\mathcal F$, let

$$\operatorname{ex}(n,\mathcal F)=\max\big\{|\mathcal H|:|V(\mathcal H)|=n\text{ and }\mathcal H\text{ is }\mathcal F\text{-free}\big\}.$$

Assume that, for all sufficiently large $n$, the extremal $n$-vertex $\mathcal F$-free $r$-graph is unique, and denote it by $\operatorname{EX}(n,\mathcal F)$. Let $N_{\mathcal F}(\mathcal G)$ be the number of labelled copies of $\mathcal F$ in $\mathcal G$, and define

$$c(n,\mathcal F)=\min\left\{N_{\mathcal F}(\operatorname{EX}(n,\mathcal F)\cup\{e\};e):e\in\binom{V(\operatorname{EX}(n,\mathcal F))}{r}\setminus\operatorname{EX}(n,\mathcal F)\right\}.$$

Mubayi's conjecture. If $\mathcal F$ is a stable non-$r$-partite $r$-graph, then, for every positive integer $q$ and all sufficiently large $n$, every $n$-vertex $r$-graph with $\operatorname{ex}(n,\mathcal F)+q$ edges contains at least

$$q\,c(n,\mathcal F)$$

copies of $\mathcal F$. This proposes that the one-edge cost above a rigid extremal construction gives the correct linear lower bound for every fixed excess $q$; the paper studies strong counterexamples, so the conjecture is not established here and its status should be checked.