  For any cohomology theory $H$ that can be factorized through (the
Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ (or
through $SH(k)$) we establish the $SH^{S^1}(k)$-functorialty (resp.
$SH(k)$-one) 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 $SH^{MGL}(k)$, 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 categories of motivic pro-spectra, and introduce
Gersten weight structures for them. Our results rely on several interesting
statements on weight structures in cocompactly cogenerated triangulated
categories and on the '$SH^+(k)$-acyclity' of primitive schemes. .
