Costa–Morini–Pasotti–Pellegrini partial-sums conjecture for abelian groups

Let $(G,+)$ be an abelian group, and let $A$ be a finite subset of $G\setminus\{0\}$. Assume that no two-element subset $\{x,-x\}$ is contained in $A$, and that

$$\sum_{a\in A}a=0.$$

For an ordering $(a_1,\ldots,a_k)$ of $A$, define partial sums by $s_j=\sum_{i=1}^j a_i$. Costa–Morini–Pasotti–Pellegrini conjecture. There exists an ordering of the elements of $A$ such that the partial sums are all distinct. The conjecture would imply that every Heffter system is simple, connecting the partial-sums problem with cyclic cycle systems. Its general status is not resolved in the supplied text.