Cichacz–Hinc conjecture on magic rectangle sets over finite abelian groups

Let $(\Gamma,+)$ be a finite abelian group, and let $\mathcal{G}$ be the set of finite abelian groups that either have odd order or contain more than one involution. For integers $m,n>1$ and $c\geq 1$, an $\mathrm{MRS}_\Gamma(m,n;c)$ is a set of $c$ $m\times n$ arrays whose entries are the elements of $\Gamma$, each appearing once, with constant row sums and constant column sums in every array. Cichacz–Hinc conjecture. An $\mathrm{MRS}_\Gamma(m,n;c)$ exists if and only if $m$ and $n$ are both even, or $\Gamma\in\mathcal{G}$ and $\{m,n\}\neq\{2\ell+1,2\}$. The construction of magic rectangle sets is generally open, although necessary and sufficient conditions for their existence are known; this conjecture proposes a complete characterization in terms of the parity of $m,n$ and the structure of $\Gamma$.