The refined threshold conjecture for expanded cliques with bounded matching number

Let $K_{\ell+1}^{(r)+}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices, let $M_{s+1}^{(r)+}$ be an $r$-uniform matching with $s+1$ edges, and let $t_r(n,\ell-1)$ denote the number of edges in the complete balanced $(\ell-1)$-partite $r$-graph on $n$ vertices. Refined threshold conjecture. For integers $\ell\geq r\geq3$, there is a constant $s_1=s_1(r,\ell)\geq \frac{\ell^2}{2}$ such that, for sufficiently large $n$, when $\binom{\ell}{2}\le s<s_1$,

$${\mathrm{ex}}_r(n,\{K_{\ell+1}^{(r)+},M_{s+1}^{(r)+}\})=\binom{\ell}{2}\cdot \binom{n-\binom{\ell}{2}}{r-1},$$

and when $s\geq s_1$,

$${\mathrm{ex}}_r(n,\{K_{\ell+1}^{(r)+},M_{s+1}^{(r)+}\})=s\cdot t_r(n,\ell-1).$$

This refines the preceding conjecture by proposing a second asymptotic regime below a threshold $s_1$; the supplied text gives no resolution.