J. Rosenberg's $\mathbb{ S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{ S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between $\partial_\theta\in T\mathbb{ S}^1$ and the normal vector field to $X\times \{ P\}$, for a fixed $P\in \mathbb{ S}^1.$
Categories: math.DG