We note first that by Proposition 2.9 (viii) the property that ${P}_{\theta}(u)$ is a uniform limit of $\theta$-psh model functions is equivalent to the property that it is a continuous function. Let $K^{\prime}/K$ be a finite normal extension and denote by $q\colon X^{\prime}\coloneqq X\otimes_{K}{K^{\prime}}\to X$ the natural projection. Let $\theta^{\prime}=q^{*}\theta\in\mathcal{Z}^{1,1}(X^{\prime})$. Then by Lemma 2.11 we have that

It follows from [Ber90, Prop. 1.3.5] that ${X^{{\mathrm{an}}}}$ is as a topological space equal to the quotient of $(X^{\prime})^{\rm an}$ by the automorphism group of $K^{\prime}/K$. We conclude that ${P}_{\theta}(u)$ is continuous if and only if ${P}_{\theta^{\prime}}(q^{*}u)$ is continuous.

Hence we can replace $K$ by a finite normal extension. Adapting the same argument as in [BFJ15, Lemma A.7] to characteristic $p$, there exists a finite normal extension $K^{\prime}/K$ and a function field $F$ of transcendence degree $d$ over $k$ with $K^{\prime}$ as completion as in Definiton 8.1 such that $X^{\prime}\coloneqq X\otimes_{K}K^{\prime}$ is the base change of a projective variety $Y$ over $F$ with $N^{1}(Y/F)_{\mathbb{Q}}\to N^{1}(X^{\prime}/K^{\prime})_{\mathbb{Q}}$ surjective. Replacing $K^{\prime}$ by $K$, we can assume that there is $Y$ as above with a surjective map

induced by the natural projection $X\to Y$. To prove continuity of ${P}_{\theta}(u)$, we may assume that the de Rham class $\{\theta\}$ is in $N^{1}(X)_{\mathbb{Q}}$ by using an approximation argument based on Proposition 2.9(vi). We conclude from surjectivity in (8.2) that there is a non-zero $m\in N$ such that $\{m\theta\}$ is induced by a line bundle $L$ with $(X,L)$ of geometric origin from a $d$-dimensional family over $k$ (in fact from $Y$). By Proposition 2.9(vii), we have ${P}_{\theta}(u)=\frac{1}{m}{P}_{m\theta}(mu)$ and hence continuity follows from Lemma 8.5. ∎