Quantum generalized Springer decomposition conjecture

Let $G$ be a connected reductive group. A $\mathtt{q}$-cuspidal datum is a $G$-conjugacy class of pairs $(L,C)$, where $L$ is an elliptic-pseudo Levi subgroup of $G$ and $C$ is a simple unipotent cuspidal ${\mathcal D_\mathtt{q}(L)}$-module. Let $Z(L)^\circ$ denote the identity component of the center of $L$. Quantum generalized Springer conjecture. The category ${\mathcal D}_q(G)\textrm{-mod}_G^{str}{G}$ has a finite block decomposition indexed by the set of $\mathtt{q}$-cuspidal data; moreover, the block corresponding to $(L,C)$ is equivalent to the category of modules for the smash product of $\mathcal D_\mathtt{q}(Z(L)^\circ)$ with a twisted group algebra of a certain finite group. This is presented as the $\mathtt{q}$-analogue of Theorem A cited in the source. The source further notes that proving it requires quantum Springer theory and parabolic induction and restriction functors.