Hilburn's open-embedding conjecture for partial Coulomb-branch resolutions

Let $G$ act on ${\bf N}$, let $\nu\colon\mathbb{C}^{\times}\to\mathsf{A}$ define a partial resolution $\widetilde{\mathcal{M}}_{C,\nu}\to\mathcal{M}_C$, and let $\gamma\colon\mathbb{C}^{\times}\to G$ be a cocharacter. Hilburn's open-embedding conjecture. There is a $\pi_1(G)^{\wedge}$-equivariant open embedding

$$j\colon\mathcal{M}_C(Z_G(\gamma),{\bf N}^{(\gamma,\nu)})\hookrightarrow\widetilde{\mathcal{M}}_{C,\nu}$$

for each $\gamma$. The conjecture is attributed to Justin Hilburn and is presented without a resolution in the source.