JSJ characterization conjecture for commensurably coHopfian hyperbolic groups

Let $\Gamma$ be a one-ended hyperbolic group that is not Fuchsian. Its JSJ decomposition is the canonical graph-of-groups decomposition over two-ended subgroups, with vertex groups including maximal hanging Fuchsian vertex groups and two-ended vertex groups. JSJ characterization conjecture. If $\Gamma$ is not commensurably coHopfian, then its JSJ decomposition contains a maximal hanging Fuchsian vertex group. Moreover, if the JSJ decomposition of $\Gamma$ only contains maximal hanging Fuchsian vertex groups and two-ended vertex groups, then $\Gamma$ is not commensurably coHopfian. This conjecture proposes a relationship between commensurable coHopficity and maximal hanging Fuchsian vertex groups; the paper presents examples and identifies these implications as open problems.