A $Γ$-magic rectangle set $\mathrm{MRS}_Γ(a, b; c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $Γ$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $ω\in Γ$ and the sum of the elements of each column is equal to a constant $δ\in Γ$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an $\mathrm{MRS}_Γ(a,b;c)$. We also generalize this problem, describing constructions of $Γ$-magic rectangle sets, whose elements are partially filled arrays.