Equivariant Hikita-Nakajima kernel conjecture for general quivers

Let $I=(I_0,I_1)$ be a finite quiver with dimension vector ${\bf v}=(v_i)_{i\in I_0}$ and framing ${\bf w}=(w_i)_{i\in I_0}$. Let $A$ act on the Nakajima quiver variety $\mathfrak{M}({\bf v},{\bf w})$, let $G_{\bf v}$ be the gauge group, and let $\mathcal{R}_{{\bf v},{\bf w}}$ be the corresponding variety of triples. Consider the surjective maps from $H^*_{A\times G_{\bf v}}(\operatorname{pt},\mathbb{C})$ to $H^*_A(\mathfrak{M}({\bf v},{\bf w}),\mathbb{C})$ and to the algebra of functions on the $\mathbb{T}$-fixed points of the Coulomb-branch spectrum, where $\mathbb{T}=(\mathbb{C}^\times)^{|I_0|}$. Equivariant Hikita-Nakajima kernel conjecture. The kernels of these two maps are equal; equivalently, they define the same closed subschemes of $\operatorname{Spec}H^*_{A\times G_{\bf v}}(\operatorname{pt},\mathbb{C})$. The Jordan-quiver case follows from the main theorem of the paper, but the assertion for general quivers remains open.