DAHA-Khovanov–Rozansky correspondence conjecture

Let $H\!D_{r,s}(b;q,t,a)$ have positive coefficients, let $\mathcal{H}=\bigoplus\mathcal{H}_{i,j,k}$ be the triply graded space for the torus knot $\{r,s\}$ and weight $b$, and let $d_n$ satisfy the stated image-kernel dimension condition and the associated spectral sequence be degenerate. Define $H\!D^{[n]}_{r,s}$ and $\widetilde{J\!D}^{n}_{r,s}$ by the reduction procedure in the source. DAHA-Khovanov–Rozansky correspondence conjecture. (i) The dimensions of $H\!D^{[n]}_{r,s}$ and $\widetilde{J\!D}^{n}_{r,s}$, computed as sums of absolute values of coefficients, coincide. (ii) $K\!R^n_{r,s}(b;q,t)$ coincides with the corresponding reduced Khovanov–Rozansky polynomial after tilde-normalization. (iii) The dimension of $\widetilde{J\!D}^{n}_{r,s}$ equals ${\text{\rm dim}}_{\mathbb C}\,_{\mathbb C}\mathcal{H}$, and the $q_{st}$-grading associated with $j$ is recovered up to an overall shift in $j$. The source explicitly says it does not conjecture that the claim is always true even under positivity, so this is a conditional proposed correspondence rather than an unconditional universal conjecture.