Equal-sum partition conjecture for groups with at least three involutions

Let $\Gamma$ be an Abelian group in $\mathcal{G}$, where $\mathcal{G}$ denotes the class used in the paper, and let $g_0$ be its identity element. Suppose that $\Gamma$ has even order $n$ and exactly $2^t-1\geq 3$ involutions. For every partition

$$n-1=r_1+r_2+\cdots+r_t$$

with $r_i\geq 3$ for $1\leq i\leq t$, and for every positive integer $t$ for which such a partition is considered, there is a partition of $\Gamma\setminus\{g_0\}$ into pairwise disjoint subsets $A_1,A_2,\ldots,A_t$ satisfying

$$|A_i|=r_i\quad\text{and}\quad\sum_{a\in A_i}a=g_0\qquad(1\leq i\leq t).$$

This extends the paper's equal-sum partition results for groups in $\mathcal{G}$; the supplied text does not state whether the conjecture has been resolved.