Cichacz's zero-sum partition conjecture for groups with one involution

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

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

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

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

for every $i\in[1,t]$. This extends the corresponding result known for cyclic groups; the paper states that the conjecture remains open.