Characterization conjecture for magic rectangle sets with exceptional dimensions

A $\Gamma$-magic rectangle set $\operatorname{MRS}_{\Gamma}(a,b;c)$ on an Abelian group $\Gamma$ of order $abc$ is a collection of $c$ arrays of size $a\times b$, whose entries are the elements of $\Gamma$, each appearing once, with constant row sums and constant column sums in every array. Let $a,b>1$. Magic rectangle set characterization conjecture. A $\Gamma$-magic rectangle set $\operatorname{MRS}_{\Gamma}(a,b;c)$ exists if and only if $a$ and $b$ are both even or $\Gamma\in\mathcal{G}$, and $\{a,b\}\ne\{2k+1,2\}$. A complete characterization is known outside the exceptional family $\{a,b\}=\{2k+1,2^{\alpha}\}$, while the cases with $\alpha>1$ remain unresolved in general; the stated criterion is intended to address the exceptional dimensions with one dimension equal to $2$.