I have already asked basically the same question here, but now I have found a way to rephrase it simply, so this new formulation might be more interesting.

Consider a union-closed family $\mathcal{F}$ of $n$ finite sets with $\mathcal{F} \not = \{ \emptyset \}$.

Let $\mathcal{H} \subset \mathcal{F}$ be the family of all sets which are (not necessarily proper) supersets of at least $\lceil n/2 \rceil$ of the sets in $\mathcal{F}$.

I conjecture that there always exists a non-empty set in $\mathcal{F}$ which is a subset of at least $| \mathcal{H} | - 1$ of the sets in $\mathcal{H}$.

Can we say something or find a counterexample for this conjecture?

I have tried many examples but couldn't find a counterexample.

Proving the conjecture should be difficult, because I believe it implies the union-closed sets conjecture, however finding a counterexample might be easier.