Humphreys' conjecture on blocks for reduced enveloping algebras

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, let ${\mathfrak g}$ be its Lie algebra, and let $\chi\in{\mathfrak g}^{*}$ be nilpotent. Let $\Lambda_\chi$ be the relevant set of weights and let $W_{\bullet}$ denote the corresponding dot-action of the Weyl group. The map $f$ associates to each block of $U_\chi({\mathfrak g})$ an element of $\Lambda_\chi/W_{\bullet}$. Humphreys' conjecture on blocks. The map $f$ is a bijection; equivalently, there is a natural bijection between the blocks of $U_\chi({\mathfrak g})$ and $\Lambda_\chi/W_{\bullet}$, so that

$$\left\vert\{\operatorname{Blocks}\, U_\chi({\mathfrak g})\}\right\vert=\left\vert\Lambda_\chi/W_{\bullet}\right\vert.$$

This conjecture parametrises the blocks of reduced enveloping algebras. The paper states that it is known under Jantzen's standard assumptions and establishes it under slightly weaker assumptions, including a new proof for type $G_2$ in characteristic $3$; the general claim is therefore not presented as resolved in the supplied text.