Hausel's nonequivariant abelianization conjecture

Let $G$ be a group with maximal torus $T$, Weyl group $W=N(T)/T$, and associated symplectic quotients $\mathfrak{M}_{\alpha,0}(G)$ and $\mathfrak{M}_{\alpha,0}(T)$. Let $\Phi_0(G)$ be the nonequivariant Kirwan map obtained from the equivariant Kirwan map by setting the equivariant parameter $x$ to zero, and let

$$e_0=\prod_{\beta\in\Delta}\beta\in\operatorname{Sym}\mathfrak t^*\cong H_T^*(T^*V),$$

where $\Delta\subseteq\mathfrak t^*$ is the set of roots of $G$ and $\operatorname{Ann}(e_0)$ denotes the ideal of classes annihilated by $e_0$. Hausel's nonequivariant abelianization conjecture. If $\Phi_0(G)$ is surjective, then there is a natural isomorphism

$$H^*(\mathfrak{M}_{\alpha,0}(G))\cong H^*(\mathfrak{M}_{\alpha,0}(T))^W\big/\operatorname{Ann}(e_0).$$

The equivariant analogue is conditional on surjectivity of the equivariant Kirwan map; this conjecture proposes the corresponding nonequivariant description of the cohomology ring.