Real monopole–Heegaard Floer homology isomorphism conjecture

Let $(Y,\iota)$ be a real three-manifold with a compatible real $\mathrm{spin^c}$ structure $\mathfrak{s}$. Let $\widetilde{\mathit{HMR}}$, $\widecheck{\mathit{HMR}}$, $\widehat{\mathit{HMR}}$, and $\overline{\mathit{HMR}}$ denote the real monopole Floer homology variants, and let $\widehat{\mathit{HFR}}$, $\mathit{HFR}^{+}$, $\mathit{HFR}^{-}$, and $\mathit{HFR}^{\infty}$ denote the corresponding real Heegaard Floer variants. Real Floer theories isomorphism conjecture. For a real rational homology three-sphere, or more generally any real three-manifold, there should be isomorphisms

$$\widetilde{\mathit{HMR}}(Y,\iota,\mathfrak{s})\cong \widehat{\mathit{HFR}}(Y,\iota,\mathfrak{s}),\qquad \widecheck{\mathit{HMR}}(Y,\iota,\mathfrak{s})\cong \mathit{HFR}^{+}(Y,\iota,\mathfrak{s}),$$ $$\widehat{\mathit{HMR}}(Y,\iota,\mathfrak{s})\cong \mathit{HFR}^{-}(Y,\iota,\mathfrak{s}),\qquad \overline{\mathit{HMR}}(Y,\iota,\mathfrak{s})\cong \mathit{HFR}^{\infty}(Y,\iota,\mathfrak{s}).$$

These should be isomorphisms of ${\mathbb F}_2$-vector spaces or modules over a suitable ring. The conjecture is presented as a real analogue of the isomorphism between ordinary monopole and Heegaard Floer homologies.