The conjectural equivalence of sutured instanton and Heegaard Floer homology

Let $(M,\gamma)$ be a balanced sutured manifold. Write $SHI(M,\gamma)$ for sutured instanton homology and $SFH(M,\gamma)$ for sutured Heegaard Floer homology. For a knot $K$ in a closed oriented $3$-manifold $Y$, write $I^{\sharp}(Y)$ for framed instanton Floer homology, $\widehat{HF}(Y)$ for the hat version of Heegaard Floer homology, $KHI(Y,K)$ for instanton knot homology, and $\widehat{HFK}(Y,K)$ for the hat version of knot Floer homology.

Sutured instanton–Heegaard Floer equivalence conjecture. For every balanced sutured manifold $(M,\gamma)$, there is an isomorphism

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

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

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

Sutured Heegaard Floer homology is known to be isomorphic to sutured monopole homology, but the relation between sutured instanton homology and the other Floer theories remains open.