The periodic tiling conjecture for finite subsets of finitely generated Abelian groups

Let $G$ be a finitely generated Abelian group and let $F$ be a finite subset of $G$. A tiling of $G$ by $F$ is a set $X\subset G$ such that

$$F\oplus X=G.$$

A tiling is periodic if its translation set is invariant under translations by a finite-index subgroup of $G$.

Periodic tiling conjecture. The tiling equation $F\oplus X=G$ is not aperiodic; equivalently, whenever $F$ tiles $G$ by translations, it admits at least one periodic tiling.

This conjecture concerns whether every translational tiling by a finite tile has a periodic realization. Its status is not specified in the source.