Altinsik–Yildiz–Keskin conjecture on gcd-closed sets and GCD–LCM divisibility

Let $S$ be a gcd-closed set, meaning that the gcd of any two elements of $S$ belongs to $S$. For $x\in S$ with $|G_S(x)|\geq 2$, say that $x$ satisfies condition $\mathcal M$ if, for every pair of distinct elements $y,z\in G_S(x)$, one has

$$[y,z]=x.$$

The set $S$ satisfies condition $\mathcal M$ if every $x\in S$ with $|G_S(x)|\geq 2$ does so. Here $(S)$ and $[S]$ denote the GCD and LCM matrices of $S$.

Altinsik–Yildiz–Keskin conjecture. If $\max_{x\in S}|G_S(x)|\geq 2$ and $S$ does not satisfy condition $\mathcal M$, then $(S)$ does not divide $[S]$.

This conjecture generalizes the earlier divisibility conjecture and is motivated by the characterization of gcd-closed sets of size $8$. The source states that the general divisibility problem remains unresolved beyond the cases already characterized.