6 problems

Let $T=\{1,\dots,n\}$ with $n>3$. Write ${\bf L}_T$ and ${\bf G}_T$ for the LCM and GCD matrices indexed by $T$, respectively, and call $\lambda$ a g-eigenvalue of ${\bf L}_T$ to…

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 $\mathcal C$.

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 $[S]$.

Zhao's conjecture. If $m\geq 4$ and $n<\binom{m}{2}+m+2$, then $(S^e)$ does not divide $[S^e]$ in $M_n(\mathbb Z)$.

Non-divisibility conjecture. If

Zhao's conjecture. If