Type-count bounds for arbitrary formulas on monadically stable classes

Prove that for every monadically stable class C and every first-order formula with tuples of object and parameter variables, the corresponding realized type system has near-linear growth in the size of the parameter set, namely O(|A|^{1+epsilon}) for every epsilon greater than zero.

Background

For formulas with two free variables, the paper cites an O(|A|^{1+epsilon}) type-count bound on monadically stable classes. It reports the conjecture that the same phenomenon holds for formulas with arbitrary tuples, paralleling the known result for nowhere dense classes.