3 problems

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 $M_n(\mathbb Z)$.