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For any cohomology theory $H$ that can be factorized through (theMorel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ (orthrough $SH(k)$) we establish the $SH^{S^1}(k)$-functorialty (resp.$SH(k)$-one) of coniveau spectral sequences for $H$. We also prove: for anyaffine 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 isalso true for any primitive $S$. Moreover, the cohomology of such an $S$ is adirect summand of the cohomology of any its open dense subscheme. This is avast generalization of the results of a previous paper. In order to prove theseresults we consider certain categories of motivic pro-spectra, and introduceGersten weight structures for them. Our results rely on several interestingstatements on weight structures in cocompactly cogenerated triangulatedcategories and on the '$SH^+(k)$-acyclity' of primitive schemes. .

Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences