Fixed-size partial-sum sequencing problem

Let $k$ be fixed, and let $A$ be a $k$-element subset of the nonzero elements of ${\mathbb Z}_n$. Fixed-size sequencing problem. Determine whether there exists a constant $N$ such that, whenever

$$n\geq N,$$

the partial-sum sequencing conjecture holds for $A$. This is a fixed-$k$, large-$n$ generalization of the main conjecture. The source states the problem but supplies no proof, value of $N$, or resolution.