The universal isolated-clique conjecture for extremal uniform hypergraphs

Let $G$ be an extremal $\ell$-uniform hypergraph for Problem 13 on $n$ vertices, and let $K_{t+1}^{\ell}$ denote the complete $\ell$-uniform hypergraph on $t+1$ vertices. An isolated copy of $K_{t+1}^{\ell}$ is a copy having no edge intersecting it and extending outside its vertex set. Universal isolated-clique conjecture. If

$$n>Ct$$

for some absolute constant $C$, then every extremal hypergraph for Problem 13 contains an isolated copy of $K_{t+1}^{\ell}$. The theorem preceding this question establishes only the existence of an extremal hypergraph with this property for a sufficiently large $n$; whether every extremal hypergraph has it remains open.