The Kervaire conjecture for free products with an infinite cyclic factor

Let $A$ be a nontrivial group, set $H=A\star\mathbb Z$, 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 closure of $w\in H$. Kervaire's 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 Kervaire conjecture in the case of a free product with an infinite cyclic factor. The paper states that the conjecture remains open in general, with results for torsion-free and finite groups.