2-Lusztig conjecture for Lie and Coxeter 2-representations

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, let $\chi\in\mathfrak{g}^*$ be its nilpotent character, and let $G$ be the associated finite component group. Let $\Theta_{\mathrm{Lie}}(\chi)$ and $\Theta_{\mathrm{Coxeter}}(\chi)$ be the two 2-representations of $G$ constructed respectively from the generic block of the reduced enveloping algebra and from the corresponding double cell of the Langlands-dual affine Weyl group. Write $X$ for their underlying decorated $G$-set and $[\mu]\in H^2(G,({\mathbb C}^{\times})^X)$ for the cohomology class. 2-Lusztig conjecture. The 2-representations

$$\Theta_{\mathrm{Lie}}(\chi)\quad\text{and}\quad\Theta_{\mathrm{Coxeter}}(\chi)$$

are isomorphic. Moreover, their cohomology class $[\mu]\in H^2(G,({\mathbb C}^{\times})^X)$ is trivial. This is described as a reformulation of a conjecture of Lusztig and as the penultimate conjecture of Gunnells, Rose and Rumynin; it compares the representation arising from modular Lie theory with the geometric Coxeter-cell construction.