Quasi-isometric rigidity conjecture for graphs of rigid free groups

Let $\Gamma$ be the fundamental group of a graph of groups with a JSJ decomposition in which all non-cyclic vertex groups are rigid free groups. The group $\Gamma$ is quasi-isometrically rigid if every finitely generated group quasi-isometric to $\Gamma$ is commensurable with $\Gamma$.

Quasi-isometric rigidity conjecture. If $\Gamma$ is the fundamental group of a graph of groups with a JSJ decomposition in which all non-cyclic vertex groups are rigid free groups, then $\Gamma$ is quasi-isometrically rigid.

This conjecture would extend the paper's quasi-isometric rigidity theorem to graphs of rigid free groups with cyclic edge groups. The authors explain that the expected approach uses pattern rigidity, trees with line patterns, and a generalization of Leighton's theorem, but that the argument is not completed here.