The Witten-style conjecture for finite-order mapping tori

Let $X$ be a smooth closed oriented $4$-manifold such that

$$H_*(X;\mathbb Z)=H_*(S^1\times S^3;\mathbb Z)\quad\text{and}\quad H_*(\widetilde X;\mathbb Q)=H_*(S^3;\mathbb Q),$$

where $\widetilde X$ is the universal abelian cover of $X$. The invariants $\lambda_{\,\rm{SW}}(X)$ and $\lambda_{\,\rm{FO}}(X)$ are respectively the Seiberg–Witten and Furuta–Ohta gauge-theoretic invariants of $X$. The Witten-style conjecture. For any $X$ satisfying these conditions,

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

This conjecture proposes equality, up to sign, between the Seiberg–Witten and Donaldson-type invariants for homology $S^1\times S^3$ manifolds. It is motivated by the Witten conjecture relating Donaldson and Seiberg–Witten theories, but the supplied text gives no resolution status.