The closure induction conjecture for union-closed families

Let $[n]$ be a finite universe, let $\mathcal F\subseteq 2^{[n]}$ be a union-closed family, and define its closure by

$$\overline{\mathcal F}=\{A\in 2^{[n]}:\mathcal F\cup\{A\}\text{ is union-closed}\}.$$

Suppose that $\overline{\mathcal F}$ satisfies Frankl's union-closed sets conjecture. Then $\mathcal F$ also satisfies Frankl's union-closed sets conjecture.

This is presented as an equivalent induction step for the union-closed sets conjecture, with density providing the proposed induction parameter. The source gives no resolution of this closure-based formulation.