  A filtration of Soergel bimodules by virtual crossing bimodules extends to
Rouquier's complexes associated with braid words. We show that these complexes
are invariant up to filtered homotopy with respect to the second Reidemeister
move, and the filtration of the triply graded link diagram homology,
constructed by Khovanov through the application of the Hochschild homology, is
invariant under Markov moves. We also prove that the homotopy equivalence of
the complexes of braid words related by the third Reidemeister move violates
filtration by at most two units.
