Poonen's twin-free structural formulation

Let $\mathcal{F}$ be a finite union-closed family of sets, and let $M$ be its largest set. Assume that for every pair of elements $a,b\in M$, the family contains a set $A$ with $|A\cap\{a,b\}|=1$. Let $x$ be the only element belonging to at least half of the sets of $\mathcal{F}$. Poonen's structural conjecture. If $|M|\geq 2$, then $\mathcal{F}$ consists of $\emptyset$ together with precisely all subsets of $M$ containing $x$; if $|M|=1$, then $\mathcal{F}=\{\{x\}\}$ or $\mathcal{F}=\{\emptyset,\{x\}\}$. This is the equivalent formulation associated with the twin-free condition and is stated as open.