The framed real monopole Floer and real Heegaard Floer isomorphism conjecture

Let $(Y,\tau,\mathbf{p})$ be a multi-based real three-manifold, let $\widetilde{\operatorname{HMR}}_*(Y,\tau,\mathbf{p})$ denote framed real monopole Floer homology, and let $\widehat{\operatorname{HFR}}_*(Y,\tau,\mathbf{p})$ denote hat real Heegaard Floer homology. Let $\tilde{\chi}$ be the conjectural tilde grading from the mod-two Euler characteristic conjecture. The framed real monopole Floer and real Heegaard Floer isomorphism conjecture. There is an isomorphism

$$\widetilde{\operatorname{HMR}}_*(Y,\tau,\mathbf{p})\cong\widehat{\operatorname{HFR}}_*(Y,\tau,\mathbf{p}).$$

This isomorphism is grading-preserving for the internal homological grading on framed real monopole Floer homology and the Maslov grading on real Heegaard Floer homology, and also for the hat grading on real Heegaard Floer homology and the conjectural tilde grading on framed real monopole Floer homology. The claim is motivated by the corresponding isomorphisms in ordinary monopole and Heegaard Floer theories, but is not proved in the paper.