The KO-width–Rosenberg index conjecture

Let $M$ be a closed spin manifold. Assume that $M$ has infinite $\mathcal{KO}$-width, and let $\alpha(M)$ denote its Rosenberg index in $\KO_*(\Cstar\pi_1M)$.

KO-width–Rosenberg index conjecture. One has

$$\alpha(M)\neq0\in\KO_*(\Cstar\pi_1M).$$

The conjecture proposes that infinite $\mathcal{KO}$-width is detected by the Rosenberg index. The source describes this as an expected relation between two index-theoretic obstructions and gives no proof or counterexample.