Zhao–Chen–Hong conjecture on power GCD–LCM divisibility and condition C

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 $\mathcal C$ if, for every pair of distinct elements $y,z\in G_S(x)$,

$$[y,z]=x\quad\text{and}\quad (y,z)\in G_S(y)\cap G_S(z).$$

The set $S$ satisfies condition $\mathcal C$ if every $x\in S$ with $|G_S(x)|\geq 2$ does so. Let $e$ be a positive integer, and let $(S^e)$ and $[S^e]$ denote the power GCD and power LCM matrices, respectively.

Zhao–Chen–Hong conjecture. If $\max_{x\in S}|G_S(x)|\geq 4$, then $[S^e]$ is divisible by $(S^e)$ in $M_n(\mathbb Z)$ if and only if $S$ satisfies condition $\mathcal C$.

The source records that the case $\max_{x\in S}|G_S(x)|=3$ has been proved, leaving the range at least $4$ as the unresolved part of the divisibility problem. This conjecture proposes a necessary and sufficient condition in that remaining range.