The symplectic Khovanov homology conjecture

Let $\kappa$ be an oriented link, represented as the closure of an $m$-stranded braid, and let $\mathit{Kh}_{\mathrm{symp}}^*(\kappa)$ denote the graded symplectic invariant defined by Lagrangian Floer cohomology. Write $\mathit{Kh}^{i,j}(\kappa)$ for Khovanov's bigraded link homology. Symplectic Khovanov homology conjecture. For every integer $k$, there is an isomorphism of graded abelian groups

$$\mathit{Kh}_{\mathrm{symp}}^k(\kappa) \cong \bigoplus_{i-j=k}\mathit{Kh}^{i,j}(\kappa).$$

Equivalently, the symplectic invariant should coincide with Khovanov homology after reversing the sign convention for the $j$-grading and collapsing the bigrading to the single grading $i-j$. The preceding theorem establishes that the symplectic construction is an oriented link invariant; the proposed identification with Khovanov homology is not proved here.