The generalized Ginzburg–Kazhdan conjecture for Coulomb branches

Let $G$ be a reductive group, let $G^\vee=\operatorname{SL}(N)$, and let $\mathscr A$ and $\mathscr B$ be the sheaves used to define

$$W^{g,b}=\operatorname{Spec} H^*_{G_{\mathcal O}}(\mathrm{Gr}_G,\mathscr A^b\otimes^!\mathscr B^g).$$

Generalized Ginzburg–Kazhdan conjecture. The variety $W^{g,b}$ is isomorphic to the Coulomb branch of the gauge theory associated with the quiver for a Riemann surface with $b$ legs and $g$ loops at the central vertex. This extends the established star-shaped case; the supplied text gives no resolution of the general statement.