Linear-density partial-sum sequencing problem

Let $A$ be a $k$-element subset of the nonzero elements of ${\mathbb Z}_n$, and let the partial-sum sequencing conjecture be the assertion that $A$ has an ordering with pairwise distinct partial sums. Linear-density sequencing problem. Find a constant $c$ such that

$$k\leq cn$$

implies that the partial-sum sequencing conjecture holds. This asks for a uniform positive-density range in which the main conjecture can be proved. The source presents it as a generalization of the preceding conjectures and gives no value of $c$ or resolution.