The instanton–unoriented knot Floer homology correspondence conjecture

Let $Y$ be a closed, oriented $3$-manifold and $K\subset Y$ a knot. Let $HFK'_2(Y,K;\mathbb{C})$ be the homology of the mapping-cone complex

$$CFK'_2(Y,K)=\operatorname{Cone}\bigl(\widehat{CFK}(Y,K)\xrightarrow{\Psi+\Phi}\widehat{CFK}(Y,K)\bigr),$$

and let $I^\sharp(Y,K;\mathbb{C})$ denote unreduced singular instanton homology with complex coefficients. Here $\widebar{K}$ is the mirror knot and $Kh(\widebar{K};\mathbb{C})$ is its Khovanov homology. The instanton–unoriented Floer conjecture. There is an isomorphism

$$HFK'_2(Y,K;\mathbb{C})\cong I^\sharp(Y,K;\mathbb{C}),$$

and a spectral sequence from $Kh(\widebar{K};\mathbb{C})$ to $HFK'_2(Y,K;\mathbb{C})$. This conjecture would relate the proposed unreduced unoriented knot Floer theory to singular instanton homology and provide a Khovanov-to-Floer spectral sequence; the supplied source gives no resolution status.