Random hypergraph removal conjecture

Let $K_{k+1}^{(k)}$ be the complete $k$-uniform hypergraph on $k+1$ vertices. Let $H$ be the random $k$-uniform hypergraph on $n$ vertices obtained by selecting each $k$-edge independently with probability $p$.

Random hypergraph removal conjecture. For every $\delta>0$ and every integer $k\geq 3$, there exist constants $\epsilon>0$ and $C$ such that, if $p\geq Cn^{-1/k}$, then with probability tending to $1$ as $n\to\infty$, every subgraph of $H$ containing at most $\epsilon p^{k+1}n^{k+1}$ copies of $K_{k+1}^{(k)}$ may be made $K_{k+1}^{(k)}$-free by removing at most $\delta pn^k$ edges.

This is the random analogue of the hypergraph removal lemma, whose dense form implies Szemerédi's theorem. The statement is known for $k=3$ but remains open for general $k$.