The asymptotic formula for cliques in a random uniform hypergraph

Let $G^{(r)}(n,p)$ be the random $r$-uniform hypergraph on $n$ vertices in which each possible hyperedge occurs independently with probability $p$, and let $X_{n,p}^{(r)}$ count its maximal complete subhypergraphs. Then

$$\mathbb E[X_{n,p}^{(r)}]=\sum_{k=1}^n \binom nk p^{\binom kr}(1-p^{\binom k{r-1}})^{n-k}.$$

Asymptotic clique-count conjecture. The expected number of cliques satisfies

$$\mathbb E[X_{n,p}^{(r)}]=\exp\left( \left(\frac{\log n}{-\log p}\right)^{\frac1{r-1}} \left( (1-\tfrac1{r!})\log n-\tfrac1{r-1}\log\log n +O(1) \right) \right).$$

This extends the problem of counting cliques in $G(n,p)$ to uniform hypergraphs. The supplied text does not state whether this asymptotic formula has been proved or remains open.