Hypergraph logarithm-free inducibility bound conjecture

For an $n$-vertex $r$-uniform hypergraph $G$, let $X_{G,k}$ be the number of hyperedges induced by a uniformly random $k$-vertex subset, and let $\operatorname{ind}_r(k,\ell)$ denote the corresponding asymptotic maximum point probability. For $0\le\ell\le\binom{k}{r}$, define

$$\ell^*=\min\mathopen{}\mathclose\bgroup\originalleft\{\ell,\binom{k}{r}-\ell\aftergroup\egroup\originalright\}.$$

Hypergraph logarithm-free inducibility bound conjecture. For any $r,k$ and any $0\le\ell\le\binom{k}{r}$, we have

$$\operatorname{ind}_r(k,\ell)=O\mathopen{}\mathclose\bgroup\originalleft(\sqrt{k^{r-1}/\ell^*}\aftergroup\egroup\originalright).$$

This is presented as the natural hypergraph generalisation of the graph conjecture. The supplied text does not resolve it; it notes additional difficulties in the sparse case.