The Freiheitssatz for proper powers in arbitrary free products

Let $\{A_\lambda\}$ be a family of groups and let $\star_\lambda A_\lambda$ be their free product. For a fixed factor $A_\lambda$, let $w$ be an element of the free product that is not conjugate to an element of $A_\lambda$, and let $m\ge2$. The proper-power Freiheitssatz conjecture. The natural map

A_\lambda\to (\star_\lambda A_\lambda)/\savebox{\@brx}{$\m@th{\langle}$}% \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}} w^m\savebox{\@brx}{$\m@th{\rangle}$}% \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}

is injective. This is a proper-power generalization of the Freiheitssatz for one-relator products. The conjecture is known for $m\ge4$ by Howie, while the paper gives new partial results reaching exponent $m=2$ for a broad class of groups.