Rosenberg's product conjecture for positive scalar curvature

Let $\Sigma$ be a closed manifold of dimension at least $3$, with $\dim\Sigma\ne4$. A positive scalar curvature metric is a Riemannian metric whose scalar curvature is everywhere positive. Rosenberg's conjecture. The manifold $\Sigma$ admits a positive scalar curvature metric if and only if $\Sigma\times{\mathbb S}^1$ does. Rosenberg found a counterexample in dimension $4$, while the conjecture remains open in all higher dimensions. The results in the paper require only the special case in which $\Sigma\times{\mathbb S}^1$ admits a warped product metric of positive scalar curvature.