Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?

An obvious case is when $Y$ itself admits a psc metric. Are there any other case? It was proved by Gromov and Lawson that any 3-manifold which contains a $K(\pi,1)$ prime factor does not admits a psc metric (another result in the paper implies that $Y$ should not be hyperbolic if $Y\times S^{1}$ admits psc), which, together with the Geometrization conjecture classifies all the 3-manifold admits psc metric. Does their proof works for $Y\times S^{1}$?

(I am most interested in the special case that $Y$ is an irreducible integer homology sphere)