Realizing wedges of Moore spaces as classifying spaces of finite semigroups

Fiedorowicz suggested that it was likely that every finite simply connected CW complex is homotopy equivalent to the classifying space of a finite semigroup. We prove that every finite wedge of simply connected Moore spaces of finitely generated abelian groups is homotopy equivalent to the classifying space of a finite semigroup. Consequently, homology groups alone cannot preclude a finite simply connected CW complex from being homotopy equivalent to the classifying space of a finite semigroup.