The HOMFLY homology differentials conjecture

Let $K$ be a knot, and suppose $\mathcal H_{i,j,k}(K)$ is a triply graded homology categorifying the reduced HOMFLY polynomial, equipped with differentials $\{d_N\}_{N\in\mathbb Z}$ satisfying the grading, anticommutativity, and symmetry axioms stated in the source. Define

$$\mathcal H^N_{p,k}(K)=\bigoplus_{iN+j=p}\mathcal H_{i,j,k}(K).$$

HOMFLY homology differentials conjecture. There is a homology theory $\mathcal H_*$ categorifying the HOMFLY polynomial, with differentials $\{d_N\}$ satisfying those three axioms, such that for all $N>0$ the homology of $(\mathcal H^N_*(K),d_N)$ is isomorphic to the $sl(N)$ Khovanov–Rozansky homology, while for $N=0$ the homology of $(\mathcal H^0_*(K),d_0)$ is isomorphic to knot Floer homology. This conjecture seeks one triply graded theory simultaneously recovering the $sl(N)$ theories and knot Floer homology; the source gives no resolution status.