Humphreys's conjecture on simple modules, component-group orbits, and left cells

Fix a regular block of $U_\chi(\mathfrak{g})$, with $\chi$ nilpotent, and let $\mathcal{S}$ be a complete set of nonisomorphic simple modules in this block. If $\chi$ corresponds to $e\in\mathfrak{g}$, let $A(e)$ be the component group associated with $e$, let $\Omega$ be the two-sided cell corresponding under Lusztig's bijection to the orbit of $e$, and let $\mathcal{L}$ be the collection of left cells in $\Omega$. Let $\Gamma$ denote the canonical left cell in $\Omega$. Humphreys's conjecture. There is a natural map

$$\varphi:\mathcal{S}\longrightarrow\mathcal{L}$$

onto $\mathcal{L}$ whose fibers are the orbits of $A(e)$ in $\mathcal{S}$; moreover, a simple module fixed by $A(e)$ maps under $\varphi$ to $\Gamma$, and this module is called canonical. This conjecture refines the proposed correspondence between simple modules in regular blocks and affine-Weyl-group left cells, with the component group accounting for possible multiplicities. The source reports only speculative support from computations in special cases and gives no resolution of the conjecture.