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

Let $\mathfrak{A}_{k|\ell}$ be the class of union-closed families used in the paper, and let $\mathfrak{c}_{k|\ell}$ denote the corresponding supremal proportion of members meeting a $k$-element subset in at least $\ell$ elements. Frankl's conjecture of order $k|\ell$. If $\mathcal{A}\in\mathfrak{A}_{k|\ell}$, then there exists $S\subseteq\bigcup\mathcal{A}$ with $|S|=k$ such that at least

$$2^{-k}\sum_{i=\ell}^k\binom{k}{i}$$

of the members $A\in\mathcal{A}$ satisfy $|A\cap S|\geq\ell$; equivalently,

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

In particular, $\mathfrak{c}_{k|1}=2^{-k}(2^k-1)$, $\mathfrak{c}_{k|k}=2^{-k}$, and $\mathfrak{c}_{2k+1|k+1}=2^{-1}$. These variants refine the original Frankl conjecture by prescribing the size and intersection threshold of the witnessing subset; their general validity is open.