Jiang's higher-order union-closed families conjecture

Let $\mathcal{F}$ be a union-closed family of sets and let

$$n=|U(\mathcal{F})|.$$

Jiang's conjecture. For every positive integer $k\leq n$, there exists a set $S\subseteq U(\mathcal{F})$ with $|S|=k$ that is contained in at least

$$2^{-k}|\mathcal{F}|$$

of the sets in $\mathcal{F}$.

For $k=1$ this reduces to Frankl's conjecture. The source attributes this generalization to Y. Jiang and provides no resolution of the full statement.