Nakajima's monopole bubbling conjecture for Coulomb branches

Let $G_c$ be the compact form of a gauge group, and let $mu$ and $\lambda$ be coweights with $\lambda$ dominant. For dominant $nu$ satisfying $mu\leqnu\leqlambda$, let $\mathcal M_C(mu,\lambda)$ be the moduli space of singular $G_c$-monopoles on $\mathbb R^3$, and let $\mathring{\mathcal M}_C(mu,nu)$ be the moduli space of genuine monopoles with magnetic charge $mu$ and monopole charge $nu$. Nakajima's monopole bubbling conjecture. The space $\mathring{\mathcal M}_C(mu,nu)$ is a symplectic leaf of $\mathcal M_C(mu,\lambda)$, and its transversal slice in $\mathcal M_C(mu,\lambda)$ is isomorphic to $\mathcal M_C(nu,mu)$. This conjecture has been proved for gauge theories defined by simple quivers using Coulomb branch techniques, while the general claim remains unresolved in the stated context.