Twisted braid homology–unipotent variety conjecture

Let $G$ be a split reductive group with Weyl group $W$, root lattice $\Phi$, and $V=\mathbf{Z}\Phi\otimes_{\mathbf{Z}}\mathbf{Q}$. Let $\mathit{Br}_W^+$ denote the positive braid monoid, and let $\mathcal{U}(\beta)$ and $\mathcal{X}(\beta\pi)$ be the varieties associated with $\beta\in\mathit{Br}_W^+$ and the positive full twist $\pi$. Twisted braid homology–unipotent variety conjecture. For any $\beta\in\mathit{Br}_W^+$, there is a homotopy equivalence

$$\mathcal{U}(\beta)(\mathbf{C})\simeq\mathcal{X}(\beta\pi)(\mathbf{C})$$

that matches the weight filtrations on their compactly supported cohomology. The preceding results establish the corresponding cohomological comparison in the settings treated in the paper, including type $A$ via Khovanov–Rozansky homology; the asserted general homotopy equivalence remains open.