Kronheimer–Mrowka's metric-independence conjecture for equivariant Seiberg–Witten Floer homology

Let $Y$ be an oriented rational homology $3$-sphere and let $\mathfrak{s}\in \operatorname{Spin}^c(Y)$ be a $\operatorname{Spin}^c$ structure. The equivariant Seiberg–Witten Floer homologies $HF_{{\rm to}}^{SW}(Y,\mathfrak{s})$, $HF_{{\rm from}}^{SW}(Y,\mathfrak{s})$, and $HF_{{\rm red}}^{SW}(Y,\mathfrak{s})$ are defined from the corresponding equivariant chain complexes.

Kronheimer–Mrowka's conjecture. These equivariant Seiberg–Witten Floer homologies are well-defined, meaning that they are independent of the particular choices of metrics and perturbations.

The claim would make the equivariant theories genuine invariants of the oriented rational homology $3$-sphere together with its $\operatorname{Spin}^c$ structure, unlike the irreducible Seiberg–Witten Floer homology, which is metric dependent. The supplied text attributes the statement to Kronheimer and Mrowka but gives no evidence of its resolution.