The Geometric Gersten conjecture for translation-like actions

Let $G$ be a finitely generated group. A group is word-hyperbolic in the sense of Gromov, and a group acts translation-like on $G$ when the action is free and each group element moves points by a uniformly bounded distance in a Cayley graph of $G$. A Baumslag–Solitar group is a group of the form $B(1,n)=\langle a,t\mid tat^{-1}=a^n\rangle$ for some nonzero integer $n$.

Geometric Gersten conjecture. If $G$ is a finitely generated group that is not word-hyperbolic, then some Baumslag–Solitar group acts translation-like on $G$.

The source presents this as a geometric version of a conjecture of Gersten proposed by Whyte and says that it would complete the characterization of finitely generated groups having a weakly aperiodic SFT. Its status is not resolved in the supplied text.