Hyperbolicity conjecture for one-relator groups without Baumslag–Solitar subgroups

Let $G=F/\left\langle\!\left\langle r\right\rangle\!\right\rangle$ be a one-relator group, where $F$ is a finitely generated free group and $r=[u,v]$ for some $u,v\in F$. Assume that $G$ contains no Baumslag–Solitar subgroups.

Hyperbolicity conjecture. Every such group $G$ is hyperbolic.

If true, this would imply that these groups are Hopfian and automatic, since hyperbolic groups have both properties. The source presents this as a conjecture motivating the preceding question; its status is not resolved here.