Higher-arity type bounds in monadically stable classes

Prove or refute that for every monadically stable graph class C and every first-order formula φ(x̄,ȳ), the number of φ-types over every finite parameter set A is bounded by c_ε|A|^{|x̄|+ε} for every ε>0.

Background

The paper gives an almost-linear type-counting bound for formulas with two free variables on monadically stable classes. It reports a conjecture that the corresponding bound extends to formulas with arbitrary tuples of object and parameter variables, analogous to the established theorem for nowhere dense classes.