Failure of gradualism under imperfect monitoring

We consider a two-player Prisoner's Dilemma type game with continuous actions, where players choose how much to contribute to a public project. This game is played infinitely many times and actions are irreversible: players cannot decrease their actions over time. While it is strictly dominant for players not to contribute in the stage game, some strictly positive level of contribution is Pareto optimal. It is known that when players perfectly observe each other's actions, cooperation can be achieved through gradual increases in contribution levels. I show that introducing an arbitrarily small amount of smooth noise in the monitoring makes cooperation impossible and players play the static Nash equilibrium of the stage-game forever.