The logarithmic Riemann–Hilbert correspondence conjecture for geometric local systems

Let $X$ be a smooth algebraic variety over a number field $E$. Fix an isomorphism $\iota:\overline{\mathbb{Q}}_p\stackrel{\sim}{\to}\mathbb{C}$ and a field homomorphism $\sigma:E\to\mathbb{C}$, and write $\sigma X=X\otimes_{E,\sigma}\mathbb{C}$. Let $v$ be the $p$-adic place of $E$ induced by $E\xrightarrow{\sigma}\mathbb{C}\xrightarrow{\iota^{-1}}\overline{\mathbb{Q}}_p$, and let $E_v$ be the corresponding completion. A $p$-adic étale local system $\mathbb{L}$ on $X$ is geometric if every stalk at a geometric point over a closed point gives a geometric $p$-adic representation in the sense of Fontaine–Mazur. For geometric $\mathbb{L}$, the de Rham construction gives a regular integrable connection $D_{\mathrm{dR}}^{\mathrm{alg}}(\mathbb{L}|_{X_{E_v}})\otimes_{E_v,\iota}\mathbb{C}$ on $\sigma X$, equipped with its decreasing filtration $\mathrm{Fil}^{\bullet}$, while the classical Riemann–Hilbert construction gives a regular integrable connection from the associated complex local system.

Logarithmic Riemann–Hilbert correspondence conjecture. The two tensor functors from the category of geometric $p$-adic étale local systems on $X$ to the category of regular integrable connections on $\sigma X$ are canonically isomorphic. Moreover,

$$\bigl(D_{\mathrm{dR}}^{\mathrm{alg}}(\mathbb{L}|_{X_{E_v}})\otimes_{E_v,\iota}\mathbb{C},\mathrm{Fil}^{\bullet}\bigr)$$

is a complex variation of Hodge structures.

This predicts compatibility between the classical and $p$-adic de Rham realizations of geometric local systems, including the Hodge-theoretic structure carried by the de Rham realization. The supplied text does not state whether the claim has been proved or remains open.