Cohomological boundedness of twisted coherent Springer sheaves

We prove that the coherent Springer sheaf and its parabolic analogues are concentrated in cohomological degree $0$, as predicted by Ben-Zvi-Chen-Helm-Nadler, Zhu, Emerton-Gee-Hellmann, Hansen, and others. More generally, we show that the universal trace functor for a mixed partial affine Hecke category is right t-exact with respect to the exotic t-structure given by Bezrukavnikov-Mirkovi\'c's noncommutative Springer resolution, and left t-exact with respect to the monoidally dual t-structure. To this end, we construct an explicit complex computing the universal trace functor for certain monoidal categories over quotient stacks.