Achar's conjecture on equivariant coherent sheaves on the nilpotent cone

Let $G$ be a complex reductive group with nilpotent cone $ilcone$, let $K_G( ilcone)$ be the Grothendieck group of $G$-equivariant coherent sheaves on $ilcone$, and let $\mathcal{N}_{o, r}$ index pairs $(e,\tau)$ consisting of a nilpotent orbit and an irreducible equivariant local system. Let ${\Lambda_+}$ index the basis of $K_G(\mathcal{N})$ described by induced representations, and let $\gamma(e,\tau)$ be the smallest maximal-length dominant weight occurring among sheaves supported on $\overline{{\mathcal O}}_e$ and restricting to the local system arising from $\tau$. Achar's conjecture. The map $\gamma$ is well-defined and bijective, and there is a basis $\{M(e,\tau)\}$ for $K_G(\mathcal{N})$, indexed by $\mathcal{N}_{o, r}$, such that $M(e,\tau)$ is supported on $\overline{{\mathcal O}}_e$, restricts to the locally free sheaf arising from $\tau$ on ${\mathcal O}_e$, and satisfies

$$\Gamma(\mathcal{N},M(e,\tau))=\pm\operatorname{Ind}_T^G\gamma(e,\tau)+\sum_{\|\mu\|^2<\|\tau\|^2}m_\mu\operatorname{Ind}_T^G\mu.$$

This conjecture describes an upper-triangular basis relating equivariant coherent sheaves on the nilpotent cone to induced $G$-modules; the source notes that it was proved for $G=GL(n,\mathbb C)$, while the general case is left open.