The homological Hikita conjecture for quiver gauge theories

Let $Q$ be a quiver with dimension vectors $\bv$ and $\bw$, flavor torus $F$, and resolved quiver variety $\tilde{\mathcal M}_Q$. Let $\mathcal M_{Q,\ff}$ be the Poisson-deformed Coulomb branch, $\mathcal A_{Q,\ff}$ its universal quantization, and let $\nu$ be the chosen cocharacter of the Hamiltonian torus. Let $B^\nu(-)$ denote the B-algebra associated with the $u$-weight decomposition.

Homological Hikita conjecture. There is an isomorphism of graded $\mathbb C[(\mathfrak t_\bv/S_\bv)\times\ff]\otimes\mathbb C[\hbar]$-algebras

$$H^*_{F\times\mathbb C^\times_\hbar}(\tilde{\mathcal M}_Q)\simeq B^\nu(\mathcal A_{Q,\ff}).$$

In particular, after specializing at $\hbar=0$ and then at $0\in\ff$, there are isomorphisms

$$H^*_F(\tilde{\mathcal M}_Q)\simeq\mathbb C[\mathcal M_{Q,\ff}^{\nu}],$$

and

$$H^*(\tilde{\mathcal M}_Q)\simeq\mathbb C[\mathcal M_Q^{\nu}].$$

This is the additive counterpart of the K-theoretic Hikita conjecture. The paper proves the conjecture in important cases, including generalized slices for ADE quivers, but the general statement is not established.