Salzborn's formulation of the union-closed sets conjecture

Let $\mathcal{F}$ be a finite normalized family, meaning a normalized union-closed family in the terminology of the source, and let $J(\mathcal{F})$ denote its family of irreducible sets. Salzborn formulation. Then

$$\exists\, I\in J(\mathcal{F}): |I|\geq \frac{|\mathcal{F}|}{2}.$$

This is a formulation of Frankl's conjecture restricted to normalized families. The paper describes it as surprisingly unfruitful and gives no resolution, so it remains open.