Poonen's twin-free union-closed family conjecture

Let $\mathcal{F}$ be a finite union-closed family of sets with $\emptyset \notin\mathcal{F}$. Elements $a$ and $b$ are twins if every set $A\in\mathcal{F}$ satisfies $|A\cap\{a,b\}|\neq 1$; the family is twin-free if it has no twins. Let $M$ be the largest set of $\mathcal{F}$. An element is abundant if it belongs to more than half of the sets of $\mathcal{F}$. Poonen's twin-free conjecture. If precisely one element is abundant, then $\mathcal{F}$ consists exactly of all subsets of $M$ containing that element. This stronger structural conjecture is stated to remain open, and the paper reports little known about partial results.