Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given by $p$). In this situation, there is a beautiful/famous result goes by the saying "Newton above Hodge". The statement is that the Newton polygon of $H^i_{crys}(\mathcal{X}_0/W(k))[\frac{1}{p}]$ lies above or on the Hodge polygon of $H^i_{dR}(\mathcal{X}_{K})$.
The proof I'm aware of goes by proving Fontaine's Crystalline conjecture (due to Faltings, and many others, then most recently Colmez--Nizioł/Bhatt--Morrow--Scholze), which implies that the Crystalline cohomology $H^i_{crys}(\mathcal{X}_0/W(k))[\frac{1}{p}]$ as a filtered $\phi$ module is weakly admissible. (Obviously tons of details need to be said with this approach.)
This sounds like a somewhat complicated "proof". Does anyone know an alternative proof?
I have seen people stating the above result and refer to either Mazur's paper [Frobenius and the Hodge filtration (estimates)] (which is a special case) or the book by Berthelot--Ogus [Notes on Crystalline Cohomology, 8.36] (which shows the Newton polygon lies on or above the Hodge polygon $\textit{of the special fiber}$). I have also come across the paper by Ogus [Frobenius and the Hodge spectral sequence] (which generalizes the Berthelot--Ogus result above). These are certainly beautiful and related results, but it seems that one cannot quite deduce the statement in the first paragraph out of them.
