The Hikita--Nakajima conjecture for symplectically dual resolutions

Let $X$ and $X^!$ be symplectically dual, and let $\mathcal{A}$ be the universal quantization of $X$. The quantization exact sequence and the equivariant cohomology exact sequence are the sequences displayed in the source. Hikita--Nakajima conjecture. There exist an isomorphism between these exact sequences and an isomorphism of graded commutative $H^2_{T^!\times\mathbb{G}_m}(\mathrm{pt})\cong k[H_2(X)][\hbar]$-algebras

$$B(\mathcal{A})\cong H^*_{T^!\times\mathbb{G}_m}(X^!;k),$$

which specializes to the Hikita isomorphism. This conjecture is the equivariant and quantized extension of the classical Hikita conjecture and supplies the parameter identifications used by the quantum Hikita conjecture; its resolution status is not specified in the source.