The non-Archimedean Hodge–${\mathcal D}$-conjecture for varieties with good reduction

Let $X$ be a projective scheme over a global field $K$. Let $p$ be a prime in ${\mathcal O}_K$, regard $X$ as a variety over the completion $K_p$, and let ${\mathcal X}$ be a model over the ring of integers ${\mathcal O}_{K_p}$ with smooth special fibre ${\mathcal X}_p$. For $m,n\geq 1$, define

$$\Sigma^{m,n}_{X}:=\{\operatorname{Ker}\{CH^{m}({\mathcal X},n-1)\longrightarrow CH^{m}(X,n-1)\}.$$

The non-Archimedean Hodge–${\mathcal D}$-conjecture. The map

$$CH^m(X,n)\otimes {\mathbb Q}\stackrel{\partial}{\longrightarrow}CH^{m-1}({\mathcal X}_p,n-1)\otimes {\mathbb Q}$$

is surjective, and $\Sigma_X^{m,n}$ is finite.

This conjecture concerns the boundary map in the localization sequence for higher Chow groups. The paper notes that it has several known special cases, while the general assertion remains open; a semistable-reduction formulation is also possible but is not treated here.