Lusztig's conjecture on generic positive-depth parabolic induction

Let $G$ be a connected reductive group over the maximal unramified extension of a non-archimedean local field, let $G_r$ be the truncated parahoric group scheme associated to a point in a torus apartment, and let $T_r\subset B_r\subset G_r$ arise from a split maximal torus and a containing Borel subgroup. For $r>0$, let $\mathcal{L}$ be a sufficiently generic multiplicative local system on $T_r$, and define

$$\operatorname{pInd}_{B_r}^{G_r}(\mathcal{L})=\pi_!f^*\mathcal{L},$$

using the maps $f$ and $\pi$ from the associated diagram. Lusztig's conjecture. The complex $\operatorname{pInd}_{B_r}^{G_r}(\mathcal{L})$ is an intersection cohomology complex on $G_r$. This predicts that sufficiently generic positive-depth character sheaves arise by parabolic induction; the paper establishes a corresponding result under its genericity hypotheses, while the conjecture is presented here in its original form.