Frankl–Gryaznov–Talebanfard's clique-counting conjecture for hypergraphs

Let $\mathcal{H}$ be an $n$-vertex $r$-graph. A clique is a subset $S\subseteq V(\mathcal H)$ such that either $|S|<r$, or $|S|\ge r$ and every $r$-subset of $S$ belongs to $\mathcal{H}$. Let $k(\mathcal{H})$ be the number of cliques, and define

$$V(x,t)=\sum_{j=0}^{t}\binom{x}{j}.$$

Frankl–Gryaznov–Talebanfard's conjecture. If $\mathcal{H}$ contains no clique of size $ell+1$, then

$$k(\mathcal H)\le V\!\left(\frac{(r-1)n}{\ell},r-1\right)^{\frac{\ell}{r-1}}.$$

Furthermore, when $r-1\mid\ell$ and $ell\mid(r-1)n$, the unique extremal case is the $n$-vertex $ell/(r-1)$-partite $r$-graph whose edge set consists of all $r$-sets intersecting each part in at most $r-1$ vertices. This conjecture gives a sharp upper bound on the total number of cliques and specifies the equality case under the stated divisibility conditions.