Frankl–Kupavskii conjecture for the Erdős–Kleitman problem

For integers $n\ge s\ge2$, define

$$e(n,s)=\max\{ |\mathcal F|:\mathcal F\subseteq2^{[n]}\text{ and }\nu(\mathcal F)<s\}.$$

For integers $m\ge1$ and $1\le\ell\le\lceil s/2\rceil$, set $n=(m+1)s-\ell$ and, for any $L\in\binom{[n]}{\ell-1}$, define

$$P(m,s,\ell;L)=\{A\subseteq[n]: |A|+|A\cap L|\ge m+1\},$$

and write $|P(m,s,\ell)|$ for its size. Frankl–Kupavskii conjecture. Under these assumptions,

$$e(n,s)=|P(m,s,\ell)|.$$

The family $P(m,s,\ell;L)$ supplies a natural lower bound because it has no $s$ pairwise disjoint members. The conjecture remains open in the stated range, with $m=3$ identified as the first unresolved value in the conjectured range.