Collusion model with imperfect public monitoring
You'll get different answers depending on assumptions, so the answer is maybe. Two papers come to mind.

For a continuous time model, see Sannikov and Skrzypacz (2007). They show that collusion is impossible with imperfect monitoring if new information arrives continuously and actions can respond quickly in accordance with new information.

A well known model using discrete time periods is Green and Porter (1984). They show that collusion is possible. Firms can sustain collusion by allowing for the possibility of price wars.