Zhao's non-divisibility conjecture for gcd-closed sets

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a gcd-closed set, and let $(S^e)$ and $[S^e]$ denote the GCD and LCM matrices associated with the relevant entrywise construction on $S$. Set

$$m=\max_{x\in S}\{|G_S(x)|\}\geq 4.$$

Zhao's conjecture. If

$$n<\binom{m}{2}+m+2,$$

then $(S^e)\nmid [S^e]$.

This conjecture extends the known small-cardinality classifications for divisibility of GCD and LCM matrices. The source gives no resolution status.