Alspach's conjecture on distinct nonzero partial sums in cyclic groups

Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$ satisfy

$$\sum_{t\in T}t\neq0.$$

An ordering of the elements of $T$ is simple when its partial sums are all distinct and non-zero. Alspach's conjecture. Every such $T$ admits a simple ordering. The conjecture would shorten some proofs concerning the existence of cycle decompositions. It is known when $|T|=v-1$, when $|T|=v-2$, and for $v\leq16$; the general case remains open.