Generalizations of Mezhirov's game semantics to predicate superintuitionistic logics and the Casari formula

Game semantics allows us to look at basic logical concepts from another side. This approach to logic has a long history, there are plenty of different types of games: provability games, semantic games, etc. And there is an interesting type of provability games called Mezhirov's game proposed by Iliya Mezhirov for intuitionistic logic of propositions and Grzegorczyk modal logic. Mezhirov's game semantics for intuitionistic logic is interesting because of its simplicity and strong connection with Kripke semantics and Kripke models. In my study, I try to generalize Mezhirov's result in two directions: to generalize to intuitionistic logic of predicates (or to some its extensions) and to the case of a connection not only between the game and tautologies of logic, but also between the game and entailment from infinite sets of formulas.