Universal monodromic tilting sheaves

We construct the universal monodromic big tilting sheaf on base affine space and calculate its endomorphisms. By formal completion, we recover Soergel’s pro-unipotent Endomorphismensatz with arbitrary field coefficients. We give a Soergel bimodules description of the universal monodromic Hecke category and deduce a conjecture of Eberhardt that uncompletes Koszul duality.