Furuta–Ohta invariant and Seiberg–Witten invariant relation

Let $X$ be a closed oriented homology-oriented smooth 4-manifold satisfying

$$H_*(X;\mathbb Z)=H_*(S^1\times S^3;\mathbb Z),\qquad H_*(\widetilde X;\mathbb Q)=H_*(S^3;\mathbb Q).$$

Let $\lambda_{\rm FO}(X)$ and $\lambda_{\rm SW}(X)$ denote the Furuta–Ohta and Mrowka–Ruberman–Saveliev invariants, respectively. Furuta–Ohta–Seiberg–Witten conjecture.

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

The supplied status evidence says this equality has been verified in a number of examples, but does not state that it is proved in full generality.