Trinh's unipotent-variety twisting conjecture

Let $G$ be a complex, connected, reductive algebraic group, let $B_+$ and $B_-$ be opposed Borel subgroups, and let $U_+$ and $U_-$ be their unipotent radicals. Let $\mathcal{U}\subseteq G$ be the variety of unipotent elements, and define

$$\Phi:U_+U_-\longrightarrow\mathcal{U},\qquad \Phi(x_+x_-)=x_+x_-x_+^{-1}.$$

For $g\in G(\mathbf{C})$, set

$$\mathcal{U}_g=\mathcal{U}\cap gB_+,\qquad \mathcal{V}_g=U_+U_-\cap gB_+,$$

and let $\Phi_g:\mathcal{V}_g\to\mathcal{U}_g$ be the restricted map. Trinh's unipotent-variety twisting conjecture. The map $\Phi_g$ defines half of a homotopy equivalence between $\mathcal{U}_g(\mathbf{C})$ and $\mathcal{V}_g(\mathbf{C})$. This would imply a weight-preserving isomorphism between their compactly supported cohomologies. The conjecture is proposed as the central geometric idea of the paper; the source also proves an equivariant analogue in type $A$, but does not establish the general homotopy-equivalence claim.