In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and converging to the sl_N Khovanov-Rozansky homology of the knot.
My question is: are there expected to be "intermediate" spectral sequences connecting sl_N homology and sl_M homology for various N and M? Less optimistically, is it known whether the total rank of the sl_N homology is increasing in N (in the "unstable" region before it reaches the HOMFLY homology)?
E.g. the d_1 differential, as constructed by Rasmussen, starts at the HOMFLY homology and converges to the Lee homology. In its original construction, though, Lee's spectral sequence started at the standard ("sl_2") Khovanov homology. Is there a way to extract the original construction from the HOMFLY version?