Positivity conjecture for Alvis–Curtis duals of Deligne–Lusztig intersection cohomology

Let ${\mathbf G}$ be a connected reductive algebraic group with Steinberg endomorphism $F$, let $G={\mathbf G}^F$, let $W$ be its Weyl group, and let $Q_w$ denote the restriction to $G$ of the Alvis–Curtis dual of the intersection-cohomology character associated with the Deligne–Lusztig variety ${\mathrm X}(w)$. Assume that $\ell\neq p$, that $\ell$ is good for ${\mathbf G}$, and that $\ell\nmid |(Z({\mathbf G})/Z({\mathbf G})^\circ)^F|$. Positivity conjecture. For all $w\in W$ and all $\lambda\in k^\times$, $(-1)^{a(w)}Q_w[\lambda]$ is the unipotent part of a projective character, where $a(w)$ is the $a$-value of the two-sided cell containing $w$. In particular, $Q_w$ should be a proper projective character when $\ell$ is not too small. Lusztig proved the corresponding characteristic-zero positivity statement, but the modular projectivity assertion is conjectural and is intended to provide information on decomposition numbers of unipotent $\ell$-blocks.