Sphere conjecture for finite-type Coxeter systems

Prove that for a generic stability condition on a 2-Calabi–Yau category associated with a finite-type Coxeter system of rank \(r\), the simplicial complex \(\Sigma_{\tau}\) is homeomorphic to the sphere of dimension \(2r-3\).

Background

For type $A_n$, the paper proves that the complex of spherical objects is piecewise-linearly homeomorphic to a sphere of dimension $2n-3$. It then proposes the corresponding dimension formula for all finite-type Coxeter systems, where the rank replaces the type-A parameter.