Furuta–Ohta and Seiberg–Witten invariant correspondence conjecture

A homology $S^1 \times S^3$ is a smooth oriented $4$-manifold with the integral homology of $S^1 \times S^3$. A $\mathbb Z[\mathbb Z]$ homology $S^1 \times S^3$ additionally has an infinite cyclic cover with the integral homology of $S^3$. Let $X$ be such a manifold with a fixed orientation and homology orientation, and let $\lambda_{\,\operatorname{FO}}(X)$ and $\lambda_{\,\operatorname{SW}}(X)$ denote its Furuta–Ohta and Seiberg–Witten invariants, respectively.

Furuta–Ohta–Seiberg–Witten correspondence conjecture.

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

The conjecture proposes that the Donaldson-theoretic Furuta–Ohta invariant agrees, up to sign, with the Seiberg–Witten invariant constructed in the paper. The authors present evidence for this relationship, but no resolution is given here.