Completeness conjecture for weighted extremal families in the Erdős–Kleitman problem

Write $n=sm+c=s(m+1)-\ell$, and for $0\le k\le m$ define

$$\mathcal H^k(m,s,\ell)=\{F\subseteq[n]\colon k|F|+|F\cap[a_k]|\ge m(k+1)\},\qquad a_k=ms-kc-1,$$

and let $\mathcal P(m,s,\ell)$ be the shifted family $\{F\subseteq[n]\colon |F|+|F\cap[\ell-1]|\ge m+1\}$. Completeness conjecture. For every $n$ and $s$ with $n=sm+c=s(m+1)-\ell$,

$$e(n,s)=\max\left\{|\mathcal H^0(m,s,\ell)|,|\mathcal H^1(m,s,\ell)|,\ldots,|\mathcal H^m(m,s,\ell)|,|\mathcal P(m,s,\ell)|\right\}.$$

Moreover, every extremal family is isomorphic to one of the candidates attaining this maximum. The conjecture proposes that these known weighted constructions form a complete list of extremal families; the source does not give a resolution.