Frankl–Kleitman-type conjecture for the non-uniform Erdős matching problem

Let $e(n,s)$ be the maximum size of a family ${\mathcal F}\subset 2^{[n]}$ with matching number $\nu({\mathcal F})<s$. Suppose that $s\ge2$, $m\ge1$, and $0<\ell\le\lceil s/2\rceil$, with

$$n=sm+s-\ell.$$

Define

$${\mathcal P}(s,m,\ell):=\bigl\{P\subset 2^{[n]}:|P|+|P\cap[\ell-1]|\ge m+1\bigr\}.$$

Frankl–Kleitman conjecture.

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

This is a proposed non-uniform analogue of the Erdős Matching Conjecture. The family on the right is known to have matching number less than $s$, but the supplied context gives no resolution of the asserted extremality statement.