The amalgamated-product Tor-vanishing conjecture

Let $A$, $B$ and $C$ be groups, and let $G=A\ast_C B$ be their amalgamated free product. Let $K\subseteq\mathbb{C}$ be a field closed under complex conjugation, and set

$$R_G:=\mathcal{R}_{K[A]}\ast_{\mathcal{R}_{K[C]}}\mathcal{R}_{K[B]}.$$

Assume that $\mathcal{R}_{K[C]}$ is a universal localization of $K[C]$. Amalgamated-product Tor-vanishing conjecture.

$$\operatorname{Tor}_1^{R_G}(\mathcal{U}(G),\mathcal{U}(G))=0.$$

The conjecture would remove the remaining homological obstruction to extending the universal-localization construction from finite edge groups to more general amalgamated products. It is proposed as an open problem in the paper.