The positive-theory conjecture for non-cyclic one-relator groups

A one-relator group is a group admitting a presentation with one defining relator. A group has trivial positive theory when every positive sentence true in some nontrivial group is true in it. The Baumslag–Solitar group is

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

Positive-theory conjecture. A non-cyclic one-relator group has trivial positive theory unless it is isomorphic to $BS(1,n)$ for some $n\in\mathbb Z$. This is the positive-theory analogue of the conjecture characterising SQ-universal one-relator groups; the source presents it as a natural conjecture, with no resolution supplied.