Co-H-spaces and almost localization

Apart from simply-connected spaces, a non simply-connected co-H-space is a typical example of a space X with a co-action of $B\pi_1(X)$ along $r^X : X \rightarrow B\pi_{1}(X)$ the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of $r^X$ (or the `almost' p-localization of X) is a fibrewise co-H-space (or an `almost' co-H-space, resp.) for every prime p. In this paper, we show that the converse statement is true, i.e., for a non simply-connected space X with a co-action of $B\pi_1(X)$ along $r^X$, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space.