On the convergence of the orthogonal spectral sequence

We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence.