Seiberg–Witten–Casson–Furuta–Ohta invariant conjecture for homology $S^1\times S^3$

Let $X$ be a smooth oriented homology-oriented $4$-manifold with the $\mathbb{Z}[\mathbb{Z}]$-homology of $S^1\times S^3$. The invariants $\lambda_{SW}(X)$ and $\lambda_{FO}(X)$ denote the Seiberg–Witten–Casson and Furuta–Ohta invariants, respectively. Seiberg–Witten–Casson–Furuta–Ohta conjecture. One has

$$\lambda_{SW}(X)=-\lambda_{FO}(X).$$

The source states that this conjecture is verified in the case where $X$ admits a free circle action. Under the supplied status evidence, the claim is treated as resolved in that setting.