Connectivity conjecture for the rigid-module and tilting-module graphs

Let $\Lambda$ be the selfinjective preprojective algebra under consideration, let $T$ be a basic maximal rigid $\Lambda$-module, and let $\operatorname{End}_\Lambda(T)$ be its endomorphism algebra. Denote by ${\mathcal T}_\Lambda$ the graph of basic maximal rigid $\Lambda$-modules and by ${\mathcal T}_{\operatorname{End}_\Lambda(T)}$ the graph of basic tilting modules over $\operatorname{End}_\Lambda(T)$, with edges given by mutation or exchange of complements. Connectivity conjecture. The graphs ${\mathcal T}_\Lambda$ and ${\mathcal T}_{\operatorname{End}_\Lambda(T)}$ are connected. Proposition shows that the former embeds as a union of connected components of the latter, so the conjecture asserts that there is only one such component in each graph; the source gives no resolution.