In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a field $k$ can be identified with $H^p(X,\mathcal{H}^q)$, where $\mathcal{H}^q$ is the Zariski sheafification of $U\mapsto H^q(U)$, for various cohomology theories, including algebraic de Rham cohomology. These terms are the same as the terms on the second page of the conjugate spectral sequence (or "second spectral sequence" of hypercohomology, and proposition 6.4 asserts that indeed the second pages are isomorphic for de Rham cohomology when $k$ is characteristic zero.
I'm wondering whether this result could also hold in positive characteristic, and if not, what could go wrong. I know that char p de Rham cohomology often has the "wrong" ranks, and isn't generally considered a Weil cohomology theory, but does it fail the Bloch-Ogus axioms given in the paper? ((2.2) in the paper suggests that at least it was not known that it should satisfy them. But maybe this was just because Hartshorne's foundational paper was all about characteristic zero.) Poincare duality should be true, at least. Maybe one needs to restrict to proper things to avoid anomalies? Or maybe work in mixed characteristic so one can compare with crystalline cohomology - that'd be ideal; a perfect world would have a good theory spread out over $\mathbb{Z}$ but that seems too much to ask for.
My motivation for asking this question is that the collapse of the conjugate spectral sequence for algebraic de Rham cohomology is, as far as I know, much more controllable in the positive characteristic case thanks to the technology of the Cartier isomorphism, so the identification of these second pages would allow one to conclude the degeneration of the coniveau spectral sequence on the second page as well.
