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. ∎