The one-relator translation-like action conjecture

Let $G$ be a nontrivial finitely generated group. A group acts translation-like on $G$ when it acts freely and every group element moves points by a uniformly bounded distance in a Cayley graph of $G$. A nontrivial one-relator group is a nontrivial group admitting a presentation with one defining relation.

One-relator action conjecture. Some nontrivial one-relator group acts translation-like on $G$.

The paper introduces this as a new conjecture intended to help address the two preceding conjectures, concerning decidable domino problems and weakly aperiodic SFTs. No resolution is given in the supplied text.