The exact-half reformulation of Poonen's conjecture

Let $\mathcal{F}$ be a union-closed family of sets, and suppose that a most frequent element of $U(\mathcal{F})$ belongs to exactly half the sets in $\mathcal{F}$. Exact-half conjecture. Then $\mathcal{F}$ must be a power set.

This is presented as a weakening of Poonen's refinement of Frankl's conjecture. The source gives no resolution, so the claim remains open.