We will present a necessary and sufficient condition (called the strong MOMO property) for uniquely ergodic measure preserving system (Z,D,m,R) to have all uniquely ergodic model of the system Möbius disjoint. In particular, we will show that Sarnak's conjecture on Möbius disjointness implies uniform convergence of ergodic averages with Möbius weights in every (topological) zero entropy system. We will also discuss the absence of the strong MOMO property in positive entropy systems.