Dade's conjecture for finite groups

Let $G$ be a finite group and let $u_\ell$ denote the $\ell$-adic valuation. For $\chi\in\operatorname{Irr}(G)$, define its $\ell$-defect by $d_\ell(\chi):=\nu_\ell(|G|)-\nu_\ell(\chi(1))$. For a non-negative integer $d$, let $\operatorname{Irr}^d(G)$ be the set of irreducible characters $\chi\in\operatorname{Irr}(G)$ with $d_\ell(\chi)=d$. Let $\mathcal{O}_G(\mathcal{S}_\ell(G))$ be the orbit space of chains of $\ell$-subgroups $\sigma=\{P_0=1<P_1<\dots<P_n\}$, let $|\sigma|:=n$ be the length of such a chain, let $G_\sigma$ be its stabiliser in $G$, and let ${\bf k}^d(G_\sigma)$ be the number of characters in $\operatorname{Irr}^d(G_\sigma)$. Dade's Conjecture. For any finite group $G$ and any positive integer $d$,

$$\sum\limits_{\sigma\in\mathcal{O}_G(\mathcal{S}_\ell(G))}(-1)^{|\sigma|}{\bf k}^d(G_\sigma)=0.$$

This is the block-free form of Dade's Conjecture, asserting an alternating-sum cancellation over chains of $\ell$-subgroups. Its status is not established by the supplied source context.