Connected-components conjecture for nongeneric wreath character varieties

Let $\mathcal{C}$ be a tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$, not necessarily generic, and suppose that the character variety $\operatorname{Ch}_{\mathcal{C}}$ is nonempty. Connected-components conjecture. The variety $\operatorname{Ch}_{\mathcal{C}}$ has two connected components if and only if every conjugacy class in $\mathcal{C}$ has connected centralizer; otherwise it is connected. This predicts the connectedness pattern beyond the generic case; the supplied text gives examples and observations but no proof of the general assertion.