KST-type lower-bound conjecture from Sidorenko exponents

Let $r\geq 2$ be an integer and let $F$ be an $(r-1)$-graph. Let $F(t)$ denote the $r$-uniform hypergraph obtained by adjoining a new vertex to every edge of $F$, and let $\operatorname{ex}(n,H)$ denote the $n$-vertex extremal number of $H$. Let $s(F)$ be the Sidorenko exponent of $F$. KST-type lower-bound conjecture. There exists a positive integer $C_F$ such that, for all $t>C_F$,

$$\operatorname{ex}(n,F(t))\geq\Omega_F\left(n^{r-\frac{1}{s(F)}}\right).$$

The conjecture asserts that the paper's general upper bound for these extremal numbers is best possible for sufficiently large $t$. It extends known lower-bound results for complete multipartite hypergraphs, but the general statement remains open.