The partial-sum sequencing conjecture for subsets of cyclic groups

Let $A=a_1,ots,a_k$ be a subset of the nonzero elements of ${\mathbb Z}_n$. For an ordering $(a_1,\ldots,a_k)$ of $A$, define the partial sums by

$$s_j=\sum_{i=1}^j a_i\qquad (1\leq j\leq k),$$

with arithmetic in ${\mathbb Z}_n$. Partial-sum sequencing conjecture. There exists an ordering of the elements of $A$ such that the partial sums are all distinct; equivalently, $s_i\ne s_j$ whenever $1\leq i<j\leq k$. This conjecture generalizes sequencing questions for cyclic groups and is motivated by cyclic cycle systems and embeddings of complete graphs on surfaces. The source reports partial results: the claim is known for $k\leq5$, for $n\leq16$, and for $k=n-1$ or $k=n-2$ via the related Alspach conjecture. The general case remains open.