Flach–Morin's local conjecture ${\bf D}_p(\mathcal{X},n)$

Fix a prime number $p$ and let $\mathcal X$ be regular, proper and flat over $\operatorname{Spec}(\mathbb{Z})$. Write $R\Gamma_{dR}(\mathcal{X}_{\mathbb{Q}_p}/\mathbb{Q}_p)/F^n$ for the derived de Rham complex modulo the Hodge filtration, and let $R\Gamma(\mathcal{X}_{\mathbb{Z}_p,et},\mathbb{Q}_p(n))$ and $R\Gamma(\mathcal{X}_{\mathbb{F}_p,eh},\mathbb{Q}_p(n))$ denote the indicated étale and eh-topology motivic complexes. Flach–Morin's conjecture ${\bf D}_p(\mathcal{X},n)$. There is an exact triangle of complexes of $\mathbb{Q}_p$-vector spaces

$$R\Gamma_{dR}(\mathcal{X}_{\mathbb{Q}_p}/\mathbb{Q}_p)/F^n[-1]\rightarrow R\Gamma(\mathcal{X}_{\mathbb{Z}_p,et},\mathbb{Q}_p(n))\rightarrow R\Gamma(\mathcal{X}_{\mathbb{F}_p,eh},\mathbb{Q}_p(n)).$$

This conjecture relates the de Rham realization of the arithmetic surface at $p$ to its étale and special-fibre motivic realizations. It is used in the study of special values of zeta functions, but its resolution status is not specified in the supplied text.