On the Same $n$-Types for the Wedges of the Eilenberg-Maclane Spaces

Given a connected CW-space $X,\ SNT(X)$ denotes the set of all homotopy types $X'']$ such that the Postnikov approximations $X^{(n)}$ and $X''^{(n)}$ are homotopy equivalent for all $n$. The main purpose of this paper is to show that the set of all the same homotopy $n$-types of the suspension of the wedges of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type of itself, i.e., $SNT(\Sigma (K(\Bbb Z, 2a_1) \vee K(\Bbb Z, 2a_2) \vee \cdots \vee K(\Bbb Z, 2a_k))) = *$ for $a_1 < a_2 < \cdots < a_k$, as a far more general conjecture than the original one of the same $n$-type posed by McGibbon and M{\o}ller (in McGibbon, C. A. and M{\o}ller, J. M., On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres, Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Lecture Notes in Math., {\bf 1509}, 1992, 285--293].)