The real monopole Floer–Seiberg–Witten Floer spectrum isomorphism

Let $K$ be a nonzero determinant link. Write $\widetilde{\operatorname{HMR}}_{\bullet}(K)$ for its reduced real monopole Floer homology and let $\operatorname{SWF}_R(K)$ denote the real Seiberg–Witten Floer spectrum. Then the real monopole Floer–Seiberg–Witten Floer spectrum conjecture. There is an isomorphism

$$\widetilde{\operatorname{HMR}}_{\bullet}(K) \cong \widetilde{H}_*(\operatorname{SWF}_R(K);\mathbb{F}_2).$$

The conjecture proposes that the algebraically defined real monopole Floer homology agrees with the reduced $\mathbb{F}_2$-homology of the real Seiberg–Witten Floer spectrum. This would relate the mapping-cone construction to the spectrum-level invariant; the source does not state whether the isomorphism is known in general.