The Lemke-Kleitman conjecture on index-one subsequences

Let $n$ be a positive integer, let $T$ be a sequence of length $n$ over the cyclic group $\mathbb{Z}_n$, and call a subsequence of $T$ index one when its index is one. Lemke-Kleitman's conjecture. Every sequence $T$ of length $n$ over $\mathbb{Z}_n$ contains an index-one subsequence. The conjecture was disproved in general by the author together with Gao, Li, Peng and Plyley, but remains open for $\mathbb{Z}_p$ when $p$ is prime.