The positive average overlap density conjecture

For a finite set $\{1,2,\ldots,n\}$, let $\mathcal{F}\subset\mathcal{P}(\{1,2,\ldots,n\})$ be a union-closed family with $\mathcal{F}\neq\{\emptyset\}$. Define

$$\operatorname{AOD}(\mathcal{F})=\mathbb{E}_{A\in\mathcal{F}\setminus\{\emptyset\}}\mathbb{E}_{B\in\mathcal{F}}\left[\frac{|A\cap B|}{|A|}\right].$$

Positive average overlap density conjecture. There exists an absolute positive constant $c>0$ such that, for every $n\in\mathbb{N}$ and every such $\mathcal{F}$, the average overlap density of $\mathcal{F}$ is at least $c$. The source presents this as the weakening that remained open after the $1/2$ conjecture was disproved.