Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be a vertical blow-up such that $\varphi=\varphi_{D}$ for some $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$. Since $\theta$ is determined by $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$, the assumption that $\varphi$ is $\theta$-psh implies that $D$ is $\pi$-nef. By Lemma 1.4 and Kleiman’s criterion [Kle66], we may find a vertical $\pi$-ample $\mathbf{Q}$-divisor $A\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ arbitrarily close to $D$. It is then clear that $\varphi_{A}$ is uniformly close to $\varphi=\varphi_{D}$ on $X$ (see the proof of Corollary 2.4). Since $A$ is $\pi$-ample we may find $m\gg 1$ such that $\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ is $\pi$-globally generated. If we set $\mathfrak{a}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ we then have $\varphi_{A}=\tfrac{1}{m}\log|\mathfrak{a}|$, which concludes the proof. ∎