Minimization version of the union-closed sets conjecture

Let $\mathcal{F}$ be a union-closed family with nonempty set collection $\mathcal{S}(\mathcal{F})$, and let $m(\mathcal{F})$ be its number of sets and $a(\mathcal{F})$ the maximum number of sets containing a single element. For any positive integer $m$, define

$$\mathscr{G}(m)=\{\mathcal{F}\mid \mathcal{F}\textup{ is a union-closed family},\ \mathcal{S}(\mathcal{F})\neq \emptyset\textup{ and }m(\mathcal{F})=m\}.$$

Minimization conjecture. For all $m\in\mathbb{N}^+$,

$$\min_{\mathcal{F}\in\mathscr{G}(m)}a(\mathcal{F})\geq\frac{m}{2}.$$

This is the companion optimization formulation of Frankl's conjecture. The paper presents it as a new conjecture with computational support and special cases, but it remains open.