The LCM-closed sets conjecture

Let $\mathcal{N}\subset\mathbb{N}^+$ be a finite LCM-closed set, meaning that $\operatorname{lcm}(m,n)\in\mathcal{N}$ whenever $m,n\in\mathcal{N}$, and suppose that $\mathcal{N}$ contains an element different from $1$. A divisor larger than $1$ is called abundant if it divides at least half of the members of $\mathcal{N}$. The LCM-closed sets conjecture. There exists a divisor larger than $1$ for a subset of at least half of the members of $\mathcal{N}$. This conjecture is equivalent to Frankl's union-closed sets conjecture and has interpretations in graph and lattice theory; it remains open because Frankl's conjecture remains open.