It follows from the comparison principle (see [GZ 2]) that

$$\int_{(\varphi<\psi)}(\omega+dd^{c}\psi)^{n}\leq\int_{(\varphi<\psi)}(\omega+dd^{c}\varphi)^{n}=\int_{(\varphi<\psi)}e^{t(\varphi-\psi)}(\omega+dd^{c}\psi)^{n}.$$

Since $e^{t(\varphi-\psi)}<1$ on $(\varphi<\psi)$, we infer $\varphi\geq\psi$ for $\nu$ almost every point, where $\nu=(\omega+dd^{c}\psi)^{n}$. Reversing the roles of $\varphi,\psi$ yields $\varphi=\psi$ for $\nu$ almost every point. Therefore $(\omega+dd^{c}\varphi)^{n}=(\omega+dd^{c}\psi)^{n}$, hence $\varphi-\psi=c$ is constant by Theorem 3.4 in [GZ 2]. Finally $c=0$ since $e^{tc}=1$ and $t>0$. ∎