On the index-conjecture on the length four minimal zero-sum sequences

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_{1}g)\cdot \mathrm{.}\mathrm{.}\mathrm{.}\cdot (n_{l}g)$ where $g\in G$ and $n_1,\cdots ,n_l\in [1,\text{ord}(g)]$, and the index $\text{ind}(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots +n_l)\mathrm{/}\text{ord}(g)$ over all possible $g\in G$ such that $⟨g⟩=G$. A conjecture says that if $G$ is finite such that $\gcd (\mathrm{\mid }G\mathrm{\mid },6)=1$, then $\text{ind}(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and at least one $n_i$ coprime to $\mathrm{\mid }G\mathrm{\mid }$.