The K-theoretic Hikita conjecture for quiver gauge theories

Let $Q$ be a quiver with dimension vectors $\bv$ and $\bw$, flavor torus $F$, and chosen cocharacter $u$ of the Hamiltonian torus acting on the resolved quiver variety $\tilde{\mathcal M}_Q$. Let $\mathcal M_Q^\times(\bv,\bw)_F$ be the K-theoretic Coulomb branch over $F$, and let $\mathcal A_{Q,F}^\times$ be its quantization. For an algebra with a $u$-weight decomposition, write $B^\nu(-)$ for its B-algebra.

K-theoretic Hikita conjecture. There is an isomorphism of algebras over $\mathbb C[(T_{\bf{v}} / S_\bv) \times F] \otimes \mathbb C[q^{\pm 1}]$

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

In particular, specializing at $q=1$ and then at $1\in F$, there are isomorphisms

$$K^{F}(\tilde{\mathcal M}_Q) \simeq \mathbb C[(\mathcal M^\times_{Q,F})^\nu],$$

and

$$K(\tilde{\mathcal M}_Q) \simeq \mathbb C[(\mathcal M^\times_Q)^\nu].$$

This is the multiplicative, or K-theoretic, analogue of the homological Hikita conjecture for quiver gauge theories. The paper establishes several cases and relations between the homological and K-theoretic versions, while the conjecture remains open in general.