The connectedness problem for cluster tilting graphs of submodule categories

Let $W$ be the Coxeter group associated with the non-Dynkin quiver, let $\Lambda$ be its completed preprojective algebra, and let $I_w$ be the ideal associated with $w\in W$. Consider the stably 2-Calabi–Yau category $\operatorname{Sub}\nolimits\Lambda/I_w$ and its cluster tilting graph, whose vertices are cluster tilting objects and whose edges correspond to mutations.

Connectedness problem. For every $w\in W$, the cluster tilting graph of $\operatorname{Sub}\nolimits\Lambda/I_w$ is connected.

The paper verifies connectedness in the displayed small examples, but leaves the general assertion as a problem. Candidate 4 is the same statement in prose and has been merged here.