Hikita–Nakajima conjecture for BFN Coulomb branches

Let $Y=T^*N\mathbin{/\mkern-6mu/\mkern-6mu/\mkern-6mu/}_\chi G$ be the Higgs branch, with the induced action of $F\times\mathbb{C}^\times$, and let $\mathcal{A}$ be the associated graded BFN algebra. Define its $B$-algebra by

$$B(\mathcal{A}):=\mathcal{A}(0)/\langle ab:a\in\mathcal{A}(-n),\ b\in\mathcal{A}(n),\ n>0\rangle.$$

The Kirwan map $H^\bullet_{\widetilde{G}\times\mathbb{C}^\times}(pt)\to B(\mathcal{A})$ is compatible with the corresponding map to $H^\bullet_{F\times\mathbb{C}^\times}(Y)$.

Hikita–Nakajima conjecture. There is an algebra isomorphism

$$H^\bullet_{F\times\mathbb{C}^\times}(Y)\cong B(\mathcal{A})$$

compatible with the map from $H^\bullet_{\widetilde{G}\times\mathbb{C}^\times}(pt)$ to both sides.

This extends conjectures of Hikita and Nakajima and relates equivariant cohomology of the Higgs branch to the degree-zero quotient of the Coulomb-branch algebra. The source notes that the relevant Kirwan maps are often surjective, but gives no resolution of this conjecture.