Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H∗(A). Fix an A∞-structure on H∗(A) making it quasi-isomorphic to A. We construct an equivalence of categories between An+1-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of An-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.