The prime-cyclic minimal-set formulation of the Lemke-Kleitman conjecture

Let $p$ be a prime, let $\mathbb{Z}_p$ be the cyclic group of order $p$, and let $\mathcal A(\mathbb{Z}_p)$ denote the set of minimal zero-sum sequences over this group. A set $\Omega\subset\mathcal A(\mathbb{Z}_p)$ is minimal when it is minimal with respect to the relevant invariant, as defined in the source. Prime-cyclic Lemke-Kleitman formulation. There exists a minimal set $\Omega\subset\mathcal A(\mathbb{Z}_p)$ such that every $V\in\Omega$ has index one. The source states that this is equivalent to the Lemke-Kleitman conjecture for prime $p$, which remains open.