The quasi-stabilization mapping-cone conjecture for link Floer complexes

Let $\bar{\mathcal H}=(\Sigma,\bar{\boldsymbol\alpha},\bar{\boldsymbol\beta},\bar{\mathbf w},\mathbf z,\bar\tau)$ be a Heegaard diagram, and let $\mathcal H=(\Sigma,\boldsymbol\alpha,\boldsymbol\beta,\mathbf w,\mathbf z,\tau)$ be its quasi-stabilization. Suppose $w_2$ is the second basepoint, apart from $w_1$, in the component of $\Sigma-\bar{\boldsymbol\alpha}$ containing the curve $\alpha_1$. Suppose $\tau^{-1}(1)=\{w_1\}$ and $\tau(w_2)=1$, so the variables corresponding to $w_1$ and $w_2$ are $U_1$ and $U_2$, respectively, and $w_1$ is the only basepoint marked by $U_1$. Quasi-stabilization mapping-cone conjecture. For suitable almost complex structures, there is an identification between the Floer complex $\mathfrak A^-(\mathcal H,\mathbf s)$ and the mapping-cone complex

$$\mathfrak A^-(\bar{\mathcal H},\mathbf s)[[U_1]]\xrightarrow{U_1-U_2}\mathfrak A^-(\bar{\mathcal H},\mathbf s)[[U_2]].$$

This predicts that quasi-stabilization is represented algebraically by a mapping cone imposing the relation between the variables associated to the two specified basepoints; the source presents it as a conjectural identification for suitable almost complex structures.