The weak 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 use the notation for $e$-local chains and blocks introduced above. For an $\ell$-block $B$ of ${\mathbf{G}}^F$, let ${\mathbf{k}}(B)$ and ${\mathbf{k}}_{\rm c}(B)$ denote the total numbers of irreducible and $e$-cuspidal characters, respectively, and let ${\mathbf{k}}(B_\sigma)$ denote the corresponding local count.

Weak e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$,

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

where $\sigma$ runs over representatives for the action of ${\mathbf{G}}^F$ on the non-trivial $e$-local chains. This is the total-character consequence proposed as an analogue of Alperin's Weight Conjecture; the source gives no resolution evidence.