Fix a model function $f$. We need to prove

$$I:=\int_{X}f\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}\leq 0$$

Choose a common determination $\mathcal{X}$ of $\varphi$ and all the $\theta_{i}$. By continuity, we may assume $\varphi=\varphi_{D}$ for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$, and each form $\theta_{i}$ is determined by a $\mathbf{Q}$-line bundle $\mathcal{L}_{i}$ on $\mathcal{X}$. Then $I=D^{2}\cdot\mathcal{L}_{1}\cdot\ldots\cdot\mathcal{L}_{n-1}$ and the result follows from [YZ10, Theorem 2.1.1 (a)]. ∎