Orientation-reversal conjecture for the Seiberg–Witten invariant

Let $X$ be a smooth spin rational homology $S^1\times S^3$ which is oriented and homology oriented, and let $-X$ denote $X$ with reversed orientation but the same homology orientation. Orientation-reversal conjecture. The Seiberg–Witten invariant satisfies

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

This conjectural antisymmetry describes the behavior of the invariant under orientation reversal. The supplied text attributes the conjectural behavior to a remark in an external reference and gives no resolution status.