Strong Frankl's conjecture of order $k|\ell$

Let $\widetilde{\mathfrak{A}}_{k|\ell}$ be the class of weakly $k|\ell$-separated union-closed families, and let $\widetilde{\mathfrak{c}}_{k|\ell}$ be the corresponding supremal proportion defined in the paper. Strong Frankl's conjecture of order $k|\ell$.

$$\widetilde{\mathfrak{c}}_{k|\ell}=2^{-k}\sum_{i=\ell}^k\binom{k}{i}.$$

This extends the order-$k|\ell$ variant from the smaller class $\mathfrak{A}_{k|\ell}$ to weakly $k|\ell$-separated families. The paper presents it as a proposed strengthening, and its general status remains open.