The e-local character-counting conjecture for finite reductive groups

Let ${\mathbf{G}}$ be a finite reductive group with Frobenius map $F$, let $\ell$ be a prime, and let $e$ be the multiplicative order of $q$ modulo $\ell$ (with the previously fixed notation). For an $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$, let ${\mathbf{k}}^d(B)$ count its irreducible characters of defect $d$, let ${\mathbf{k}}_{\rm c}^d(B)$ count its $e$-cuspidal characters of defect $d$, and let ${\mathbf{k}}^d(B_\sigma)$ count the corresponding characters for the stabiliser of an $e$-local chain $\sigma$.

The e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$,

$${\mathbf{k}}^d(B)={\mathbf{k}}_{\rm c}^d(B)+\sum_{\sigma}(-1)^{|\sigma|+1}{\mathbf{k}}^d(B_\sigma),$$

where $\sigma$ runs over representatives for the action of ${\mathbf{G}}^F$ on the non-trivial descending chains of $e$-split Levi subgroups. This is proposed as a geometric, $e$-local analogue of Dade's character-counting conjecture for finite reductive groups; its resolution status is not specified in the source.