The contractibility conjecture for stability spaces of acyclic quivers

Let $Q$ be a finite connected acyclic quiver. Write $\operatorname{Stab}(\mathcal{D}^b(Q))$ for the space of stability conditions on $\mathcal{D}^b(Q)$, and let $\mathrm{Toss}(\mathcal{D}^b(Q))$ denote the set of totally semistable stability conditions on $\mathcal{D}^b(Q)$. Contractibility conjecture. The space $\operatorname{Stab}(\mathcal{D}^b(Q))$ contracts to $\mathrm{Toss}(\mathcal{D}^b(Q))$, the space $\mathrm{Toss}(\mathcal{D}^b(Q))$ is contractible, and consequently $\operatorname{Stab}(\mathcal{D}^b(Q))$ is contractible.

This is a categorical analogue of the classical $K(\pi,1)$-conjecture for hyperplane arrangements. Contractibility is known for stability spaces associated with Dynkin quivers and related Calabi--Yau completions, while the stated result for arbitrary finite connected acyclic quivers is presented as a direction for further research.