By Proposition 2.9(vi), we may assume that $\theta\in\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}$. By Proposition 2.9(vii), we may replace $L$ by a positive tensor power and $\theta$ by the corresponding multiple, and so we may assume that $L$ is very ample. In the notation of Definition 8.1, the assumption that $(X,L)$ is of geometric origin means that $L$ is the pull-back of a line bundle $L^{\prime}$ on the projective variety $Y$ over $K^{\prime}=k(B)$. It follows easily from [Har77, Prop. III.9.3] and [EGAIV, Prop. 2.7.1(xii)] that $L^{\prime}$ is very ample. Then $L^{\prime}$ extends to a very ample line bundle on a projective $R$-model of $Y$ for the discrete valuation ring $R=\mathcal{O}_{B,b}$ from Definition 8.1. By base change to ${K^{\circ}}$, we conclude that there is a closed $(1,1)$-form $\theta^{\prime}$ on ${X^{{\mathrm{an}}}}$ with de Rham class $\{\theta^{\prime}\}=\{\theta\}$ such that $(X,\theta^{\prime})$ is of geometric origin from a $d$-dimensional family over $k$.

By the $dd^{c}$-lemma in [BFJ16a, Thm. 4.3] (see also the second author’s thesis [Jel16, Thm. 4.2.7] for generalizations) and using the rationality assumption on $\theta$ from the beginning of the proof, there is $v\in{\mathscr{D}}(X)$ such that $\theta^{\prime}=\theta+dd^{c}v$. It follows from Proposition 2.9(iv) that

By Lemma 8.4, the function ${P}_{\theta^{\prime}}(u-v)$ is a uniform limit of $\theta^{\prime}$-psh functions. Adding $v$, we get the claim for ${P}_{\theta}(u)$. ∎