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A Riemannian manifold is said to have infinite $\mathcal{KO}$-width if itadmits an isometric immersion of an arbitrarily wide Riemannian band whoseinward boundary has non-trivial higher index. In this paper we prove that if aclosed spin manifold has inifinite $\mathcal{KO}$-width, then its Rosenbergindex does not vanish. This gives a positive answer to a conjecture by R.Zeidler. We also prove its `multi-dimensional' generalization; if a closed spinmanifold admit an isometric immersion of an arbitrarily wide cube-like domainwhose lowest dimensional corner has non-trivial higher index, then theRosenberg index of $M$ does not vanish.

Band width and the Rosenberg index