Let $k$ be a finite extension of $\mathbb{Q}_p$. In [this][1] paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says: > ...it should be possible to deduce (log-)crystalline comparison theorems... 1. What should be the correct statements of the more refined Hodge theoretic comparison theorems in the context of rigid-analytic varieties? 2. Are there any mathematical difficulties with proving these statements, or did Scholze just not feel like proving them? 3. Is there any subsequent work aiming to prove these statements? EDIT: in general, I have the following question (which may be trivial, and I only have it become of my ignorance). What is "the reduction $\mathrm{mod} \, p$" of a non-algebraic smooth proper rigid-analytic variety over $\mathbb{Q}_p$? Does it see the non-algebraicity? [1]: https://arxiv.org/abs/1205.3463