Nonexistence of infinitely many phase transitions for surface diffeomorphisms

Let $S$ be a compact surface, let $f\in \operatorname{Diff}^{1}(S)$ be a diffeomorphism, and let the potential be Holder continuous. A phase transition is a failure of analyticity or differentiability of the pressure function associated to the potential. Infinite phase-transition conjecture. There is no diffeomorphism $f\in \operatorname{Diff}^{1}(S)$ such that $f$ has infinitely many phase transitions with respect to a Holder continuous potential. The conjecture is motivated by examples with arbitrarily large finite numbers of phase transitions, while infinite phase transitions are expected not to occur in this differentiable setting.