Inversion of adjunction for arithmetic log pairs

Let $C=\operatorname{Spec}(\mathcal{O}_{K})$ be the spectrum of the ring of integers of a number field, let $\pi\colon\mathcal{X}\to C$ be as in the preceding notation, and let $\mathcal{X}_{c}=\pi^{-1}(c)$ for a closed point $c\in C$. Let $D$ be an effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor whose support contains no component of $\mathcal{X}_{c}$, and let $t\geq 0$. Inversion of adjunction. Each of the following equivalences holds: $(\mathcal{X},\mathcal{X}_{c}+tD)$ is log-canonical in a neighborhood of $\mathcal{X}_{c}$ if and only if $(\mathcal{X}_{c},D|_{\mathcal{X}_{c}})$ is geometrically semi-log-canonical, meaning semi-log-canonical after base change to an algebraic closure of $\kappa(c)$; and $(\mathcal{X},\mathcal{X}_{c}+tD)$ is purely log terminal in a neighborhood of $\mathcal{X}_{c}$ if and only if $(\mathcal{X}_{c},D|_{\mathcal{X}_{c}})$ is log-terminal. This is proposed because inversion of adjunction in the arithmetic setting is not established, motivating the formulation via log pairs; no resolution status is supplied.