The ungraded derived equivalence for perverse coherent sheaves on the nilpotent cone

Let $G$) be a simply connected semisimple algebraic group over an algebraically closed field of good characteristic, and let $\mathcal{N}$ denote the nilpotent variety in the Lie algebra of $G$. Write $D^{\mathrm{b}}\mathsf{PCoh}^G(\mathcal{N})$ for the bounded derived category of $G$-equivariant perverse coherent sheaves on $\mathcal{N}$, and $D^{\mathrm{b}}\mathsf{Coh}^G(\mathcal{N})$ for the bounded derived category of $G$-equivariant coherent sheaves on $\mathcal{N}$. Ungraded derived-equivalence conjecture. One should have

$$D^{\mathrm{b}}\mathsf{PCoh}^G(\mathcal{N}) \cong D^{\mathrm{b}}\mathsf{Coh}^G(\mathcal{N}).$$

The analogous graded equivalence with a $\mathbb{G}_{\mathrm{m}}$-action is proved in the paper, but the methods used there do not establish this ungraded statement; its status is therefore open in the supplied source.