Hong's divisibility conjecture for gcd-closed sets satisfying condition G

Let $a$ and $b$ be positive integers with $a\mid b$, and let $S$ be a finite gcd-closed set of positive integers. For each $x\in S$, let $G_S(x)$ denote the set of greatest-type divisors of $x$ in $S$. Two distinct greatest-type divisors $y_1,y_2$ of $x$ satisfy condition $\mathcal G$ if $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$. The set $S$ satisfies condition $\mathcal G$ if every $x\in S$ either has $|G_S(x)|\leq 1$, or has $|G_S(x)|\geq 2$ and every two distinct greatest-type divisors of $x$ satisfy condition $\mathcal G$. Let $[S^a]$ and $[S^b]$ denote the corresponding power LCM matrices, and let $M_{|S|}(\mathbb Z)$ be the ring of $|S|\times|S|$ integer matrices. Hong's conjecture. If $S$ is gcd-closed and satisfies condition $\mathcal G$, then $[S^a]$ divides $[S^b]$ in $M_{|S|}(\mathbb Z)$; equivalently, there is a matrix $B\in M_{|S|}(\mathbb Z)$ such that $[S^b]=B[S^a]$ or $[S^b]=[S^a]B$. Hong's conjecture extends the known result for FC sets, since every FC set is gcd-closed and satisfies condition $\mathcal G$, but its general case remains open.