The dual GCD-closed sets conjecture

Let $\mathcal{N}\subset\mathbb{N}^+$ be a finite GCD-closed set, meaning that $\operatorname{gcd}(m,n)\in\mathcal{N}$ whenever $m,n\in\mathcal{N}$, and suppose that $\mathcal{N}$ contains at least two elements. A prime power is a number of the form $p^k$ for a prime $p$ and a positive integer $k$. The dual GCD-closed sets conjecture. One of the members of $\mathcal{N}$ has a prime power that is not a prime power of more than half of the members of $\mathcal{N}$. This conjecture is the dual equivalent of the LCM-closed sets conjecture, hence of Frankl's union-closed sets conjecture; it remains open.