Ponomarenko's index conjecture for minimal zero-sum sequences

Let $G$ be the cyclic group of order $n$, and let $S$ be a minimal zero-sum sequence over $G$ of length $4$. The index $\operatorname{ind}(S)$ is the smallest number in

$$\left\{\frac{1}{n}\sum_{j=1}^{k}(ga_j)_n:g\in(\mathbb{Z}/n)^*\right\},$$

where $(x)_n$ is the least non-negative integer congruent to $x$ modulo $n$. The Index Conjecture. If $\gcd(n,6)=1$, then every minimal zero-sum sequence $S$ over $G$ of length $4$ has

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

The conjecture concerns the pairs $(k,n)$ for which every minimal zero-sum sequence of length $k$ over a cyclic group of order $n$ has index $1$. The cases $k\leq 3$ and $k>n/2+1$ are known, while the case $k=4$ is subtle; Ponomarenko verified this conjecture for $n\leq 1000$.