Coherence conjecture for affine Springer fiber sheaves

The construction associates to each $gamma\bin g(cK)$ a ${cC}^*$-equivariant quasi-coherent sheaf $cF_gamma$ on $cmathfrak C_{ccheck G}$. Coherence conjecture. For any regular semisimple $gamma\bin g(cK)$, the sheaf $cF_gamma$ is coherent:

$$cF_\gamma\bin goperatorname{Coh}_{{cC}^*}(cmathfrak C_{ccheck G}).$$

Quasi-coherence follows from the construction, whereas coherence is the stronger expected property and has numerical consequences for affine Springer fiber cohomology.