Erdős–Kleitman matching conjecture for set families

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}\subset 2^{[n]}$ with matching number $\nu(\mathcal{F})<s$. For $1\leq i\leq k$, let

$$\mathcal{A}_i^{(k)}(n,s)=\Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}.$$

Let $\mathcal{P}(m,s,\ell)$ be the family defined for $n=sm+s-\ell$ by

$$\mathcal{P}(m,s,\ell)=\bigl\{P\subset [n]: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.$$

Erdős–Kleitman matching conjecture. Suppose that $s\ge 2$, $m\ge 1$, and $n=sm+s-\ell$ for some integer $0<\ell\leq\lceil s/2\rceil$. Then

$$e(sm+s-\ell,s)=|\mathcal{P}(m,s,\ell)|.$$

The conjecture concerns the extremal size of set families avoiding an $s$-matching. It is known in several cases, including $\ell=2$, $m=1$, $s\geq \ell m+3\ell+3$, and $m=2$, while the general assertion remains open.