Let $k$ be a finite extension of $\mathbb{Q}_p$. In this 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...
- What should be the correct statements of the more refined Hodge theoretic comparison theorems in the context of rigid-analytic varieties?
- Are there any mathematical difficulties with proving these statements, or did Scholze just not feel like proving them?
- Is there any subsequent work aiming to prove these statements?