The Index Conjecture à la Shen et al. for coprime-term sequences

Let $G$ be an additive cyclic group of order $n$, and let $(y)_n$ denote the least nonnegative representative of $y$ modulo $n$. For a sequence $S=(a_1)\dots(a_k)$ over $G$, define

$$\operatorname{ind}(S)=\min\left\{\frac{\sum_{i=1}^{k}(ga_i)_n}{n}:g\in G^*\right\},$$

where $G^*$ is the set of integers less than and coprime to $n$. A sequence is minimal zero-sum if it sums to zero and has no proper, nontrivial zero-sum subsequence.

The Index Conjecture à la Shen et al. For $\gcd(n,6)=1$, let $S=(a_1)(a_2)(a_3)(a_4)$ be a minimal zero-sum sequence over $G$. Suppose $\gcd(n,a_i)=1$ for all $i$. Then

$$\operatorname{ind}(S)=1.$$

Shen et al. proved that establishing this coprime-term case suffices to prove the Index Conjecture. Thus this is a sufficient reduction of the main conjecture rather than a separate independent target; the supplied status evidence identifies the reduction as proven.