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. ∎

Every ${K^{\circ}}$-model of a projective variety $X$ is dominated by a projective ${K^{\circ}}$-model [Gub03, Proposition 10.5]. Hence we may assume that ${\mathscr{X}}_{0}$ is projective. There is $m_{1}\in{\mathbb{N}}$ and a closed immersion of $X$ into ${\mathbb{P}}_{K}^{N}$ such that $L^{\otimes m_{1}}={\mathcal{O}}_{{\mathbb{P}}_{K}^{N}}(1)|_{X}$. Then the closure of $X$ in ${\mathbb{P}}_{K^{\circ}}^{N}$ is a ${K^{\circ}}$-model ${\mathscr{X}}_{1}$ of $X$ which has an ample line bundle ${\mathscr{L}}_{1}$ such that ${\mathscr{L}}_{1}|_{X}=L^{\otimes m_{1}}$. Then the closure of the diagonal in ${\mathscr{X}}_{0}\times_{K^{\circ}}{\mathscr{X}}_{1}$ is a projective ${K^{\circ}}$-model ${\mathscr{X}}$ of $X$ and the canonical projection $p_{1}:{\mathscr{X}}\to{\mathscr{X}}_{1}$ is a projective morphism, hence there is a closed immersion of ${\mathscr{X}}$ into a projective space ${\mathbb{P}}_{{\mathscr{X}}_{1}}^{k}$ over ${\mathscr{X}}_{1}$. Let ${\mathscr{E}}$ be the restriction of ${\mathcal{O}}_{{\mathbb{P}}_{{\mathscr{X}}_{1}}^{k}}(1)$ to ${\mathscr{X}}$. Since ${\mathscr{E}}$ is relatively ample with respect to $p_{1}$ and since ${\mathscr{L}}_{1}$ is an ample line bundle on ${\mathscr{X}}_{1}$, there is $m_{2}\in{\mathbb{N}}$ such that ${\mathscr{L}}:=p_{1}^{*}({\mathscr{L}}_{1})^{\otimes m_{2}}\otimes{\mathscr{E}}$ is ample on ${\mathscr{X}}$. Then ${\mathscr{L}}$ is a ${K^{\circ}}$-model of $L^{\otimes m}$ for $m:=m_{1}m_{2}$. ∎

Since ${\rm Spec}({K^{\circ}})$ is affine, ampleness is the same as relatively ample. It remains to check that the restriction of $\omega+{\varepsilon}\theta$ to the special fibre is ample (see [Gro66, 9.6.4 and 9.6.5]). The ample cone on the special fiber is the interior of the nef cone. This proves immediately the claim. ∎

We note first that restriction gives a canonical injective homomorphism $N^{1}({\mathscr{X}}/S)\to N^{1}({\mathscr{X}}_{s})$ and the ample part of $N^{1}({\mathscr{X}}/S)$ is the preimage of the ample part of $N^{1}({\mathscr{X}}_{s})$ (see the proof of Proposition 4.12). By assumption, $\theta$ can be represented by $c_{1}({\mathscr{L}})=\sum_{i}\lambda_{i}c_{1}({\mathscr{L}}_{i})$ with line bundles ${\mathscr{L}}_{i}$ and $\lambda_{i}\in{\mathbb{R}}$. We recall that the isomorphism classes of ${K^{\circ}}$-models of $X$ form a directed set and that any ${K^{\circ}}$-model of the projective variety $X$ is dominated by a projective ${K^{\circ}}$-model. So we may assume that all ${\mathscr{L}}_{i}$ live on a common projective model ${\mathscr{X}}$. We approximate the real numbers $\lambda_{i}$ by sufficiently close rational numbers $\lambda_{i}^{\prime}$. Then the restriction $L^{\prime}$ of ${\mathscr{L}}^{\prime}:=\bigotimes_{i}{\mathscr{L}}_{i}^{\prime\otimes\lambda_{i}^{\prime}}$ to the generic fibre $X$ is a ${\mathbb{Q}}$-line bundle which is sufficiently close to $L$ in $N^{1}(X)$. Since the ample cone in $N^{1}(X)$ is open, we may assume that $L^{\prime}$ is ample as well. By Proposition 4.11, we may assume that $L^{\prime}$ admits an ample extension ${\mathscr{H}}^{\prime}\in{\rm Pic}({\mathscr{X}})_{\mathbb{Q}}$. Let $\varphi^{\prime}$ be the model function corresponding to ${\mathscr{H}}^{\prime}\otimes({\mathscr{L}}^{\prime})^{-1}$. Let now $\theta^{\prime}$ be the closed $(1,1)$-form on $X$ represented by ${\mathscr{L}}^{\prime}$. Since $dd^{c}\varphi^{\prime}+\theta^{\prime}$ is represented by $c_{1}({\mathscr{H}}^{\prime})$, we conclude that $dd^{c}\varphi^{\prime}+\theta^{\prime}$ is ${\mathscr{X}}$-positive. Since the ample cone of ${\mathscr{X}}_{s}$ is open and since the restrictions of ${\mathscr{L}},{\mathscr{L}}^{\prime}$ to the special fibre ${\mathscr{X}}_{s}$ are sufficiently close, it follows from our remark at the beginning that $c_{1}({\mathscr{H}}^{\prime})+c_{1}({\mathscr{L}})-c_{1}({\mathscr{L}}^{\prime})$ is ${\mathbb{R}}$-ample. Since $dd^{c}\varphi^{\prime}+\theta$ is represented by $c_{1}({\mathscr{H}}^{\prime})+c_{1}({\mathscr{L}})-c_{1}({\mathscr{L}}^{\prime})$, we see that $dd^{c}\varphi^{\prime}+\theta$ is ${\mathscr{X}}$-positive.

Now let $\theta$ be semipositive. Since a function in ${\mathscr{D}}(X)$ is continuous on ${X^{\rm an}}$, it is bounded and hence $c:=\sup_{X^{\rm an}}\varphi$ is bounded. We may replace $\varphi^{\prime}$ by $\varphi^{\prime}-c$ without changing $dd^{c}\varphi^{\prime}$. Since $c$ is in the value group of the algebraic closure of $K$, this is still a model function and hence we may assume $\varphi^{\prime}\leq 0$. Since the sum of a nef and an ample ${\mathbb{R}}$-line bundle remains ${\mathbb{R}}$-ample (as we can check that on the special fibre, see the proof of Proposition 4.12), we know that

is also ${\mathscr{X}}$-positive for all $0<{\varepsilon}\leq 1$. Using a rational ${\varepsilon}$ sufficiently close to $0$, we get the claim. ∎