Non-divisibility conjecture for gcd-closed sets failing condition M

Let $S$ be a gcd-closed set, and let $G_S(x)$ denote the set of greatest-type divisors of $x$ in $S$. Let condition $\mathfrak{M}$ be the condition defined in the source.

Non-divisibility conjecture. If

$$\max_{x\in S}\{|G_S(x)|\}\geq 2$$

and $S$ does not satisfy condition $\mathfrak{M}$, then

$$(S)\nmid [S].$$

This conjecture generalizes Zhao's conjecture by proposing that failure of condition $\mathfrak{M}$ itself prevents divisibility. The source gives no resolution status.