Tannenbaum's zero-sum partition conjecture for Abelian groups

Let $\Gamma$ be a finite Abelian group of order $m$, and let $I(\Gamma)$ be its set of involutions. Assume $|I(\Gamma)|>1$, and put

$$R=\Gamma\setminus\bigl(I(\Gamma)\cup\{0\}\bigr).$$

Tannenbaum's zero-sum partition conjecture. For every positive integer $t$ and every integer partition $\{m_i\}_{i=1}^t$ of $m-1$, with $m_i\geq 2$ for every $i\in[1,|R|/2]$ and $m_i\geq 3$ for every $i\in[|R|/2+1,t]$, there is a subset partition $\{S_i\}_{i=1}^t$ of $\Gamma^*=\Gamma\setminus\{0\}$ such that $|S_i|=m_i$ and

$$\sum_{s\in S_i}s=0$$

for every $i\in[1,t]$. Cichacz and Suchan disproved the conjecture in general; it holds in the cases covered by the cited $2$-ZSPP and elementary $2$-group theorems.