The clique-and-hub conjecture for the hypergraph variational problem

Let $H$ be a fixed hypergraph, and let $p=p(n)$ satisfy $n^{-c_H}\ll p\ll 1$. A clique is the collection of all $r$-tuples contained in a specified prefix $S$ of $[n]$, while a hub is the collection of all $r$-tuples intersecting such a prefix. An edge-weighted $r$-graph arises from planting one of these objects when its planted edges have weight $1$ and all other edges have weight $p$.

Clique-and-hub conjecture. The variational problem is asymptotically optimized by planting a clique and a hub.

The conjecture extends the graph case, where cliques and hubs describe the asymptotically optimal constructions. It is true for graphs and, as stated in the source, also when $H$ is a clique, but it fails in general for hypergraphs.