Fontaine–Jannsen semistable conjecture

Let $K$ be the underlying $p$-adic field, let $F$ be the coefficient field of Hyodo–Kato cohomology, and let $X$ be a proper, log-smooth, fine and saturated $\mathcal{O}_K^{\times}$-log-scheme with Cartier type reduction. Let $H^i_{\operatorname{\acute{e}t}}(X_{\overline{K},\operatorname{tr}},\mathbf Q_p)$ denote the $i$-th étale cohomology, $H^i_{\operatorname{HK}}(X)$ the Hyodo–Kato cohomology, $H^i_{\operatorname{dR}}(X_K)$ the de Rham cohomology, and let $\mathbf B_{\operatorname{st}}$ and $\mathbf B_{\operatorname{dR}}$ be the semistable and de Rham period rings. Fontaine–Jannsen's semistable conjecture. There exists a natural $\mathbf B_{\operatorname{st}}$-linear, Galois-equivariant period isomorphism

$$\alpha_i:H^i_{\operatorname{\acute{e}t}}(X_{\overline{K},\operatorname{tr}},\mathbf Q_p)\otimes_{\mathbf Q_p}\mathbf B_{\operatorname{st}}\stackrel{\sim}{\to}H^i_{\operatorname{HK}}(X)\otimes_F\mathbf B_{\operatorname{st}}$$

that preserves the Frobenius and monodromy operators and whose extension to $\mathbf B_{\operatorname{dR}}$ induces a filtered isomorphism

$$\alpha_i:H^i_{\operatorname{\acute{e}t}}(X_{\overline{K},\operatorname{tr}},\mathbf Q_p)\otimes_{\mathbf Q_p}\mathbf B_{\operatorname{dR}}\stackrel{\sim}{\to}H^i_{\operatorname{dR}}(X_K)\otimes_K\mathbf B_{\operatorname{dR}}.$$

This is the comparison conjecture relating étale, Hyodo–Kato, and de Rham cohomology for proper semistable logarithmic schemes; the supplied text does not state whether it has been resolved, so its database status is left open.