The torsion-free one-relator product conjecture

Let $A$ and $B$ be torsion-free groups, set $H=A\star B$, and let \savebox{\@brx}{\m@th{\langle}}% \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}} w\savebox{\@brx}{\m@th{\rangle}}% \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}} be the normal subgroup generated by $w\in H$. The torsion-free one-relator product conjecture. The quotient

H/\savebox{\@brx}{$\m@th{\langle}$}% \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}} w\savebox{\@brx}{$\m@th{\rangle}$}% \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}

is nontrivial for every $w\in H$. This is the two-factor torsion-free case of the nontriviality problem for one-relator products. It remains open in general; the special case with one factor equal to $\mathbb Z$ follows from Klyachko's theorem.