Prethick subsets in partitions of groups

A subset $S$ of a group $G$ is called thick if, for any finite subset $F$ of $G$, there exists $g\in G$ such that $Fg\subseteq S$, and $k$-prethick, $k\in \mathbb{N}$ if there exists a subset $K$ of $G$ such that $|K|=k$ and $KS$ is thick. For every finite partition $\mathcal{P}$ of $G$, at least one cell of $\mathcal{P}$ is $k$-prethick for some $k\in \mathbb{N}$. We show that if an infinite group $G$ is either Abelian, or countable locally finite, or countable residually finite then, for each $k\in \mathbb{N}$, $G$ can be partitioned in two not $k$-prethick subsets.