Cichacz's 4-ZSPP conjecture for Abelian groups with more than one involution

Let $\Gamma$ be a finite Abelian group, and let $I(\Gamma)$ denote its set of involutions. A finite Abelian group has 4-ZSPP if every integer partition of $|\Gamma|-1$ whose parts are all at least $4$ can be realized by a partition of $\Gamma^*=\Gamma\setminus\{0\}$ into zero-sum subsets of the corresponding sizes. Cichacz's 4-ZSPP conjecture. If $|I(\Gamma)|>1$, then $\Gamma$ has $3$-ZSPP. The surrounding text records later results proving the stronger assertion that every such group has $4$-ZSPP, so the stated conjecture is solved.