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