The Kervaire–Laudenbach conjecture

Let $A$ be a group, let $H=A\star\mathbb Z$, and let $p:H\to\mathbb Z$ be the standard projection to the $\mathbb Z$ factor. For $w\in H$, write \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}} for its normal closure. Kervaire–Laudenbach's conjecture. If $p(w)\ne0$, then the natural map

A\to 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 injective. This is a central Freiheitssatz-type conjecture and remains open in general. It is known for torsion-free $A$ when $p(w)=1$ by Klyachko's theorem, and for finite, hence residually finite, $A$ by Gerstenhaber–Rothaus.