Cui–Hu conjecture on two abundant elements

Let $\mathcal{F}$ be a finite family of sets that is union-closed, meaning that the union of any two members belongs to $\mathcal{F}$. Suppose that every set in $\mathcal{F}$ has size at least $2$. Cui–Hu conjecture. There exist at least two elements such that each belongs to more than half of the sets in $\mathcal{F}$. This stronger assertion would imply Frankl's union-closed sets conjecture. It remains open according to the source, which presents it as a conjecture by Cui and Hu.