The corresponding statement for admissible formal schemes is proven in [GRW14, Proposition A.7] and hence applies to the formal completion $\hat{{\mathscr{X}}}$ of ${\mathscr{X}}$ and its Cartier divisor $\hat{D}$ given by pull-back of $D$. By the formal GAGA-principle proved in this non-noetherian situation by Fujiwara–Kato in [FK13, Theorem I.10.1.2], the Cartier divisor $D$ is effective if and only $\hat{D}$ is an effective Cartier divisor on $\hat{{\mathscr{X}}}$. Since $D$ and $\hat{D}$ determine the same model function, we get the claim. ∎