K(π,1) conjecture for the stability-condition and hyperplane-complement spaces

Let $\mathbb W$ be the Coxeter group, let $\Upsilon_{\operatorname{reg}}$ be the regular locus of the associated complexified parameter space, and let $\operatorname{Stab}^\dagger_{\mathcal C}(\mathcal D)$ be the distinguished connected component of equivariant stability conditions.

K(π,1) conjecture. The space $\operatorname{Stab}^\dagger_{\mathcal C}(\mathcal D)$ is contractible; consequently, both $\Upsilon_{\operatorname{reg}}/\mathbb W$ and $\Upsilon_{\operatorname{reg}}$ are $K(\pi,1)$ spaces.

This conjecture is motivated by analogous expectations for Bridgeland stability conditions. The paper's covering-space theorem gives the stated relationship among these spaces, but does not establish contractibility in general; the finite-Coxeter case follows from the known $K(\pi,1)$ theorem.