Frankl–Kupavskii conjecture for the shifted family $\mathcal P(m,s,\ell)$

Let $n=s(m+1)-\ell$ and define

$$\mathcal P(m,s,\ell)=\{F\subseteq[n]\colon |F|+|F\cap[\ell-1]|\ge m+1\}.$$

For $s\ge2$, $m\ge1$, and $1\le\ell\le\lceil s/2\rceil$, this family is a weighted construction with matching number less than $s$. Frankl–Kupavskii conjecture. Suppose that $s\ge2$, $m\ge1$, and $n=s(m+1)-\ell$ for some $\ell$ with $1\le\ell\le\lceil s/2\rceil$. Then

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

The source reports proofs in several ranges, including asymptotic ranges of $\ell$, but does not state a complete resolution.