Crystalline comparison for rigid-analytic varieties 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?


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?
 A: The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.
Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the \'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.
One reference is the paper by Tan-Tong (https://arxiv.org/abs/1510.05543), settling the above result over unramified bases. They follow the Faltings-Scholze's method, build on previous work of Andreatta-Iovita (https://arxiv.org/abs/1212.3813), and also prove a result for non-trivial coefficient systems.
A reference for the general statement is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.
The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.
