Finite-generation conjecture for affine Springer fiber homology

For a regular semisimple $\gamma$, define the graded module

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

It is a module over the graded algebra $\oplus_{d=0}^{cinfty}{}_0\mathcal A_d$ and therefore defines a quasi-coherent sheaf $cF_\gamma$ on $\operatorname{Proj}\bigoplus_{d=0}^{cinfty}{}_0\mathcal A_d$. Finite-generation conjecture. The module $cmathbf F_\gamma$ is finitely generated and the sheaf $cF_\gamma$ is coherent. The paper proves this in a special $GL_n$ case, while the general assertion remains open.