Dunfield–Gukov–Rasmussen conjecture on triply graded knot homology

Let $K$ be a knot with a triply graded homology theory $\mathcal{H}_{i,j,k}(K)$, and define

$$\mathcal{P}(K)(a,q,t)=\sum a^{i}q^{j}t^{k}\dim \mathcal{H}_{i,j,k}(K),$$

with

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

Dunfield–Gukov–Rasmussen conjecture. There exists a homology theory with the stated Euler-characteristic, differential, and symmetry properties such that for all $N>1$ the homology of $(\mathcal{H}^{N}_{*}(K),d_N)$ is isomorphic to the $sl(N)$ Khovanov–Rozansky homology, for $N=0$ the homology of $(\mathcal{H}^{0}_{*}(K),d_0)$ is isomorphic to Heegaard–Floer knot homology, and the homology of $d_1$ is one-dimensional. This proposes a unified triply graded theory whose specializations recover several knot homology theories; the source does not provide a resolution status.