Lee's lower-bound conjecture for apex partite hypergraphs

Let $d\ge 2$ and let $\mathcal{H}$ be a $(d-1)$-partite $(d-1)$-uniform hypergraph. For $k\in\mathbb{N}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices, each having $\mathcal{H}$ as its common link graph. Let $s(\mathcal{H})$ be the Sidorenko exponent of $\mathcal{H}$. Lee's conjecture. There exists a constant $C=C(\mathcal{H})$ such that for all $k\ge C$,

$$\operatorname{ex}(n,\mathcal{H}(k))=\Omega_{\mathcal{H}}\!\left(n^{\,d-\frac{1}{s(\mathcal{H})}}\right).$$

Lee proved the matching upper bound $O_{\mathcal{H},k}(n^{d-1/s(\mathcal{H})})$; the conjecture asserts that this upper-bound exponent is best possible for all sufficiently large $k$.