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 ntypes 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(Σ(K(ℤ, 2a₁) ∨ K(ℤ, 2a₂) ∨∙∙∙∨ K(ℤ, 2a_k))) = * for a₁ < a₂ < ∙∙∙ < a_k, as a far more general conjecture than the original one of the same n-type posed by McGibbon and Møller (in [McGibbon, C. A. and Møller, 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., 1509, 1992, 285–293].)