Dunfield–Gukov–Rasmussen symmetry conjecture for reduced Khovanov–Rozansky homology

Let $K$ be a knot and let $\overline{\mathrm{HHH}}(K)=\bigoplus \overline{\mathrm{HHH}}_{i,j,k}(K)$ be its reduced triply graded homology, where $i$ is the $a$-grading, $j$ is the quantum grading, and $k$ is the homological grading. Dunfield–Gukov–Rasmussen's symmetry conjecture.

$$\dim\overline{\mathrm{HHH}}_{i,-2j,k}(K)=\dim\overline{\mathrm{HHH}}_{i,2j,k+2j}(K).$$

This symmetry was conjectured by Dunfield, Gukov, and Rasmussen and verified in numerous examples; the paper proves it by showing that the operator $F_2$ satisfies a hard Lefschetz property. It was also related to results concerning compactified Jacobians, Hilbert schemes of points on the plane, and rational Cherednik algebras.