Let $\mathcal{X}^{\prime}$ be a determination of $\theta$ and let $\mathcal{L}^{\prime}\in\Pic(\mathcal{X}^{\prime})_{\mathbf{R}}$ be a representative of $\theta$. The assumption implies that the $\mathbf{R}$-line bundle $L:=\mathcal{L}^{\prime}|_{\mathcal{X}_{K}}$ is ample. By Corollary 1.5 we may thus assume that $\mathcal{X}^{\prime}$ has been chosen so that $L$ admits an ample extension $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{R}}$ for each model $\mathcal{X}$ dominating $\mathcal{X}^{\prime}$. If $\pi:\mathcal{X}\to\mathcal{X}^{\prime}$ denotes the corresponding vertical blow-up then $\mathcal{L}-\pi^{*}\mathcal{L}^{\prime}=D$ for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{R}}$, and $\varphi=\varphi_{D}$ is a model function such that $\theta+dd^{c}\varphi$ is $\mathcal{X}$-positive.

Now suppose $\theta$ is semipositive and pick $\mathcal{X}$, $\varphi$ as above. Upon replacing $\varphi$ by $\varphi-\sup_{X}\varphi$ we may assume that $\varphi\leq 0$. Then the closed $(1,1)$-form

is also $\mathcal{X}$-positive for each $0<\varepsilon<1$, completing the proof since $\varphi$ is bounded. ∎