The S^1-stability conjecture for positive scalar curvature

Let $M$ be a closed connected $n$-manifold with $n>4$.

$S^1$-stability conjecture. The manifold $M$ admits a metric of positive scalar curvature if and only if its product with the circle does:

$$M\text{ admits a positive-scalar-curvature metric}\iff M\times S^1\text{ admits one}.$$

The paper uses this conjecture to relate positive scalar curvature on manifolds with abelian fundamental groups to the case of $M\times S^1$. It is stated as an unresolved conjecture.