The infinite $\mathcal{KO}$-width index conjecture

Let $M$ be a closed spin manifold. Its Rosenberg index is the index class associated with the spin Dirac operator, and $M$ is said to have infinite $\mathcal{KO}$-width when its $\mathcal{KO}$-width is unbounded. The infinite $\mathcal{KO}$-width index conjecture. Every closed spin manifold of infinite $\mathcal{KO}$-width has non-vanishing Rosenberg index.

This conjecture is motivated by the stable Gromov–Lawson–Rosenberg conjecture and the obstruction from infinite $\mathcal{KO}$-width to stable positive scalar curvature; it remains open in general.