Dudas–Malle projectivity conjecture for intersection cohomology characters

Let $G$ be a finite reductive group with Weyl group $W$, let $F$ be the relevant Frobenius endomorphism, and let $\boldsymbol{T}$ be an $F$-stable torus. For $w otin W$, let $Q_w$ denote the relevant character, let $D_G$ denote Alvis–Curtis duality, and let $\overline{\boldsymbol{X}(w)}$ be the closure of the Deligne–Lusztig variety. Write $IH^\bullet(\overline{\boldsymbol{X}(w)},k(\boldsymbol{T}^{wF})_\ell)$ for its intersection cohomology with the indicated coefficient module. Dudas–Malle conjecture. For all $w\in W$, there is a sign $\varepsilon_w=\pm1$ such that

$$Q_w=\varepsilon_w D_G\bigl([IH^\bullet(\overline{\boldsymbol{X}(w)},k(\boldsymbol{T}^{wF})_\ell)]\bigr)$$

is the character of a non-virtual projective module. This conjecture proposes a geometric construction of projective modules from intersection cohomology and would provide the projectives needed to control modular decomposition matrices. The source presents it as a proposed strategy, without reporting a proof or disproof.