Wise's cubulation conjecture for torsion-free one-relator groups

A torsion-free one-relator group is a group admitting a presentation with one relator. For integers $m,n$ with $m\neq\pm n$, let

$$BS(m,n)=\langle a,t\mid t^{-1}a^m t=a^n\rangle.$$

A group acts freely on a CAT(0) cube complex if it admits a free action by cubical isometries on a CAT(0) cube complex.

Wise's conjecture. Every torsion-free one-relator group with no subgroup isomorphic to $BS(m,n)$, for $m\neq\pm n$, acts freely on a CAT(0) cube complex.

The conjecture proposes that Baumslag–Solitar subgroups are the obstruction to this form of nonpositive-curvature geometry for torsion-free one-relator groups. Such subgroups contain distorted cyclic subgroups and force exponential Dehn-function growth, while the conjecture predicts free CAT(0) cubical actions whenever they are absent. The source does not provide evidence resolving this conjecture.