Protasov–Slobodianiuk conjecture on partitions of infinite groups

Let $G$ be an infinite group of cardinality $\varkappa$, and let $cov(A)=\min\{|X|:X\subseteq G,\ G=XA\}$ denote the covering number of a subset $A\subseteq G$. A partition into countably many cells is a decomposition

$$G=\bigcup\limits_{n<\omega}A_n.$$

Protasov–Slobodianiuk conjecture. Every infinite group $G$ of cardinality $\varkappa$ has a partition as above such that

$$cov(A_nA_n^{-1})=\varkappa$$

for each $n\in\omega$.

The conjecture is confirmed for groups of regular cardinality and for some groups of arbitrary cardinality, including Abelian groups; the general case remains open.