The Berzukavnikov's equivalence using Kirillov models

The Berzukavnikov equivalence is stated for reductive groups over (closure) finite fields. One of the reasons (correct me if I am wrong) is the concept of Whittaker sheaves, which again boils down to Artin-Schreier sheaves, something that "only exists" for finite fields. In this paper, let's say over the Betti setting, Gaitsgory proposed the Kirillov model as a substitute for Whittaker sheaves and claimed that this is enough for all practical purposes. I am wondering whether there is reference for such "Betti"-Berzukavnikov equivalence using this Kirillov model?