Fontaine's crystalline comparison conjecture

Let $K$ be a complete discretely valued $p$-adic field of characteristic $0$ with ring of integers $\mathcal{O}_K$ and perfect residue field $k$, and let $X$ be a smooth, proper scheme over $\mathcal{O}_K$. Write $X_s$ for its special fiber, let $\overline{K}$ be an algebraic closure of $K$, and let $\mathrm{B}_{\mathrm{crys}}$ be Fontaine's crystalline period ring. Fontaine's $\mathrm{C}_{\mathrm{crys}}$-conjecture. There is a natural isomorphism of $\mathrm{B}_{\mathrm{crys}}$-modules

$$\operatorname{H}^i(X_{\mathrm{\acute et}, \overline{K}}, \mathbf{Z}_p) \otimes_{\mathbf{Z}_p} \mathrm{B}_{\mathrm{crys}} \simeq \operatorname{H}^i(X_s/W(k)_{\mathrm{crys}})\otimes_{W(k)} \mathrm{B}_{\mathrm{crys}},$$

compatible with Galois actions, Frobenius structures, and filtrations. This is Fontaine's predicted comparison between étale cohomology of the generic fiber and crystalline cohomology of the special fiber; the paper's abstract states that it proves the conjecture in the relative setting, for general coefficients and allowing ramified base fields.