The instanton–Heegaard Floer correspondence conjecture

Let $(M,\gamma)$ be a balanced sutured manifold. Write $SHI$ for sutured instanton Floer homology, $SFH$ for sutured Heegaard Floer homology, $I^{\sharp}$ for framed instanton Floer homology, $\widehat{HF}$ for the hat version of Heegaard Floer homology, $KHI$ for instanton knot homology, and $\widehat{HFK}$ for Heegaard knot Floer homology. Instanton–Heegaard Floer correspondence conjecture. There are isomorphisms

$$SHI(M,\gamma)\cong SFH(M,\gamma)\otimes \mathbb{C}.$$

In particular, for a knot $K$ in a closed $3$-manifold $Y$, there are isomorphisms

$$I^{\sharp}(Y)\cong \widehat{HF}(Y)\otimes\mathbb{C}\quad\text{and}\quad KHI(Y,K)\cong \widehat{HFK}(Y,K)\otimes\mathbb{C}.$$

This would connect instanton Floer homology with the sutured, Heegaard, and knot Floer theories, which are otherwise known to agree with one another in several settings. The source states that this conjecture remains open.