  Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written
in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and
$n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ is
defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all
possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$
is finite such that $\gcd(|G|,6)=1$, then $\ind(S)=1$ for every minimal
zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$
is reduced and the (A1) condition is satisfied(see [19]).
