Sidorenko conjecture for linear hypergraphs containing an expanded clique

Let $F$ be an $r$-uniform hypergraph and let $k$ be the uniformity of the clique $K_{k+1}^k$. Distinct edges of $F$ satisfy the $k$-linear condition when $|e\cap f|<k$ for every pair of distinct edges $e,f\in F$. Let $\mathrm{E}^r(K_{k+1}^k)$ be the $r$-uniform expansion of the complete $k$-graph on $k+1$ vertices. Linear hypergraph conjecture. If $|e\cap f|<k$ for any distinct $e,f\in F$ and $F$ contains $\mathrm{E}^r(K_{k+1}^k)$ as a subgraph, then $F$ is not Sidorenko. The source notes that the case $k=2$ is known, while the conjectured extension to higher uniformities remains open.