Nagel's conjecture on multiple frequent elements in union-closed families

Let $\mathcal{F}$ be a union-closed set family, and let its ground set have size at least $k$. An element is said to appear in a set when it belongs to that set.

Nagel's conjecture. There are $k$ elements, each belonging to at least

$$\frac{1}{2^{k-1}+1}|\mathcal{F}|$$

many sets in $\mathcal{F}$.

This strengthens the Union-Closed Sets Conjecture, also known as Frankl's conjecture, which is the case $k=1$. Das and Wu proved the claim for $k\geq 3$ and for $k=2$ when $|\mathcal{F}|$ lies outside a particular range; the remaining cases for $k=2$ are analysed via linear programming in the paper.