Reimer's condition conjecture

Let $[n]=\{1,\ldots,n\}$, let ${\mathcal A}\subseteq 2^{[n]}$, and suppose ${\mathcal A}$ satisfies Condition 1: there is a filter ${\mathcal F}\subseteq 2^{[n]}$ and a bijection $A\mapsto F_A$ from ${\mathcal A}$ to ${\mathcal F}$ such that $A\subseteq F_A$ for every $A\in{\mathcal A}$ and $[A,F_A]\cap[B,F_B]=\emptyset$ for distinct $A,B\in{\mathcal A}$, where $[A,B]=\{C:A\subseteq C\subseteq B\}$. Reimer's condition conjecture. There is an element $x\in[n]$ that belongs to at least half the sets of ${\mathcal A}$. The conjecture asks whether Reimer's sufficient condition alone implies the conclusion of the union-closed sets conjecture. The paper's abstract states that this implication is false, so the conjecture is refuted.