Suppose I have a singular manifold $f: S^n \times S^n \times \dots \times S^n \rightarrow X$ such that the restriction of $f$ to the wedge sum $S^n \vee S^n \vee \dots \vee S^n$ is constant. This implies $f$ factors through $f':S^n \wedge S^n \dots \wedge S^n =S^{kn} \rightarrow X$. Of course since both $S^n \times S^n \times \dots \times S^n$ and $S^{kn}$ are null bordant, there are many bordisms between the two.
Is it the case that $f$ and $f'$ are bordant as singular manifolds? I ask because in Watanabe's paper "On Kontsevich's Characteristic Classes for Higher Dimensional Sphere Bundles II" in the proof of the main theorem (6.1), Watanabe uses this to conclude something about the image of the Hurewicz map for oriented bordism.