Finite-generation conjecture for Coulomb-branch affine Springer homology

Let $G$ be a reductive group, let $\mathcal A_G=\bigoplus_{d=0}^{\infty}{}_0\mathcal A_d$ be the graded Coulomb branch algebra, and for $\gamma\in\mathfrak g$ define

$$F_{\gamma}=\bigoplus_{k=0}^{\infty}H_*(\mathrm{Sp}_{t^k\gamma}).$$

Let $\mathcal F_{\gamma}$ be the corresponding quasi-coherent sheaf on $\operatorname{Proj}\bigoplus_{d=0}^{\infty}{}_0\mathcal A_d$. Finite-generation conjecture. The module $F_{\gamma}$ is finitely generated and the sheaf $\mathcal F_{\gamma}$ is coherent. The assertion is proved in the paper for $G=\mathrm{GL}_3$ and the specified diagonal $\gamma$, but is open in general.