The mathfrak{gl}(k) KhovanovRozansky categorification conjecture

Let $kge 2$, let $\beta\in \mathfrak{Br}_n$ be a braid, and let $L(\beta)$ denote its link closure. Write $\mathrm{H}_{0|k}(\beta)$ for the homology constructed in the paper and $\mathrm{H}_{\mathfrak{gl}(k)}^{\mathrm{KhR}}(L(\beta))$ for KhovanovRozansky homology. The $\mathfrak{gl}(k)$ categorification conjecture. For every such braid,

$$\mathrm{H}_{0|k}(\beta)=\mathrm{H}_{\mathfrak{gl}(k)}^{\mathrm{KhR}}(L(\beta)).$$

The conjecture proposes that the homology for the purely odd case categorifies the $\mathfrak{gl}(k)$ link invariant in the same way as KhovanovRozansky homology. The source points to the relationship with the Soergel category and related KhovanovRozansky categories as a route toward a proof; no resolution is given here.