Poonen's strict-majority conjecture for union-closed families

Let $\mathcal{F}$ be a union-closed family of sets. A power set is a family of the form $\mathcal{P}(X)$ for some set $X$. Poonen's conjecture. Unless $\mathcal{F}$ is a power set, it contains an element that appears in strictly more than half of its sets.

This strengthens Frankl's conjecture by requiring a strict majority outside the power-set case. It is attributed in the source to Poonen and remains open.