Strong union-closed sets conjecture

Let $\mathcal{F}$ be a reduced union-closed family on a set $S$. Strong union-closed sets conjecture. The average cardinality of members of $\mathcal{F}$ is not greater than twice the average cardinality of intersections of ordered pairs of members of $\mathcal{F}$.

The source notes that this is stronger than Frankl's conjecture and reports no counterexamples, but does not establish it.