The weakly aperiodic SFT characterization for finitely generated groups

Let $G$ be a finitely generated group. A weakly aperiodic SFT is a nonempty subshift of finite type over $G$ having no point with trivial stabilizer, while a group is virtually cyclic if it has a cyclic subgroup of finite index.

Weak-aperiodicity conjecture. $G$ has no weakly aperiodic SFT if and only if it is virtually cyclic.

The paper identifies virtually cyclic groups as the only groups for which it can prove that no aperiodic SFT exists, while many non-virtually-cyclic groups are shown to admit weakly aperiodic SFTs. The claimed converse remains open in the source.