Common elements in finite LCM closed sets.

Question

Let $X$ be a finite set of square-free integers. Suppose that whenever $a, b \in X$, we have $lcm(a, b) \in X$. Does it follow that there exists $x \in X$ that divides at least half of elements of $X$?

Answer 1

In the form in which you pose the question, the answer is negative. A counterexample is afforded by $X=\{2\cdot3\cdot5,3\cdot5\cdot7,5\cdot7\cdot2,7\cdot2\cdot3,2\cdot3\cdot5\cdot7\}$.

If instead you only demand that some $n\gt1$ divides at least half the elements of $X$, this yields an arithmetic version of the union-closed sets conjecture, which is an open problem.