  For any cohomology theory $H$ that can be factorized through (the
Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ we
establish the $SH^{S^1}(k)$-functoriality of coniveau spectral sequences for
$H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin
complex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or
$DM(k)$, then this is also true for any primitive $S$. Moreover, the cohomology
of such an $S$ is a direct summand of the cohomology of any its open dense
subscheme. This is a vast generalization of the results of a previous paper.
  In order to prove these results we consider certain triangulated categories
of motivic pro-spectra, and introduce Gersten weight structures for them. We
study in detail the notions of cohomological dimensions of scheme associated to
various categories of motivic pro-spectra; they are defined in terms of the
corresponding weight structures and count the number of non-zero Nisnevich
cohomology for sheaves in the hearts of orthogonal "homotopy" $t$-structures.
