A conjecture on partitions of groups

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\inω}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\inω$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.

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