Rational homotopy groups and Koszul algebras

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring H*(Y, Q) is a k-rescaling of H*(X,Q). If H*(X, Q) is a Koszul algebra, then the graded Lie algebra pi(*) (OmegaY) circle times Q is the k-rescaling of gr(*) (pi(1) X) circle times Q. If Y is a formal space, then the converse holds, and Y is coformal. Furthermore, if X is formal, with Koszul cohomology algebra, there exist filtered group isomorphisms between the Malcev completion of pi(1) X, the completion of [OmegaS(2k+1), OmegaY], and the Milnor-Moore group of coalgebra maps from H-* (OmegaS(2k+1),Q) to H-* (OmegaY, Q).