  We determine the local equivalence class of the Seiberg-Witten Floer stable
homotopy type of a spin rational homology 3-sphere $Y$ embedded into a spin
rational homology $S^{1} \times S^{3}$ with a positive scalar curvature metric
so that $Y$ generates the third homology. The main tool of the proof is a
relative Bauer-Furuta-type invariant on a periodic-end 4-manifold. As a
consequence, we give obstructions to positive scalar curvature metrics on spin
rational homology $S^{1} \times S^{3}$, typically described as the coincidence
of various Fr{\o}yshov-type invariants. This coincidence also yields
alternative proofs of two known obstructions by Jianfeng Lin and by the authors
for the same class of 4-manifolds.
