Supersaturation conjecture for r-partite hypergraphs with Sidorenko gap

For an $r$-partite $r$-graph $F$, let $s(F)$ be its Sidorenko gap, defined by

$$s(F):=\sup\left\{s:\exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\right\}.$$

Assume $r\geq 3$ and

$$\operatorname{ex}(F,n)=O(n^{r-\alpha})$$

for some $0<\alpha<r-1$. The hypergraph supersaturation conjecture. There exists a positive constant $c$ such that, if $G$ is an $n$-vertex $r$-graph with $e(G)\geq cn^{\alpha}$, then $G$ contains at least

$$n^{v(F)-o(1)}\left(\frac{e(G)}{n^r}\right)^{e(F)+s(F)}$$

copies of $F$.

This conjecture is motivated by the failure of the direct supersaturation analogue for non-Sidorenko hypergraphs and is intended to incorporate the Sidorenko gap. The statement is presented as an open direction in the paper.