Conjecture on the matching-restricted Turán number of Berge cliques

Let $H_{\ell+1}^r$ be the Berge clique appearing in the source, let $M_{s+1}^r$ denote a matching of size $s+1$, and let $\mathcal{G}(n,\ell,s,r)$ be the hypergraph defined earlier in the paper. For integers satisfying

$$\ell \ge r \ge 3,\qquad s \ge \binom{\ell}{2},$$

and for sufficiently large $n$, consider $r$-graphs avoiding both $H_{\ell+1}^r$ and $M_{s+1}^r$.

The matching-restricted Berge-clique conjecture.

$$\operatorname{ex}_r(n, \{H_{\ell+1}^r, M_{s+1}^r\})=s\cdot t_{r-1}(n-s,\ell-1).$$

In particular, $\mathcal{G}(n,\ell,s,r)$ is the unique extremal hypergraph.

The conjecture extends matching-restricted Turán problems for hypergraphs and proposes that, once $s\ge\binom{\ell}{2}$, the previously defined construction remains extremal. The supplied text gives a counterexample to this construction in the range $\ell-1\le s\le\binom{\ell}{2}-1$, but does not report a resolution of the stated conjecture.