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

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

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

The conjecture identifies the periodic-end Seiberg–Witten invariant with the Furuta–Ohta invariant up to sign. It is verified in the product case and in the examples discussed in the paper, while the general statement remains open.