Pick $\varphi_{0}\in\PSH(X,\theta)\cap\mathcal{D}(X)$. Upon replacing $\theta$, $\varphi_{i}$, and $\psi$ with $\theta+dd^{c}\varphi_{0}$, $\varphi_{i}-\varphi_{0}$ and $\psi-\varphi_{0}$ respectively, we may assume that $\theta$ is semipositive and that $\varphi_{i}\leq 0$ for all $i$. Adding a constant to $\psi$ we may also assume $\sup_{X}\psi=0$. Set $M:=\max_{i}\sup|\varphi_{i}|$. First assume that $\psi$ is also bounded. We claim that $\int-\psi\mu$ is bounded by a constant depending only on $M$ (but not on $\sup_{X}|\psi|$). Integrating by parts we have

$$0\leq\int(-\psi)\mu=\int(-\psi)\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})\\ +\int(-\varphi_{1})(\theta+dd^{c}\psi)\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})+\int\varphi_{1}\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n}).$$

Here the second to last integral is bounded by $M\{\theta\}^{n}$, while the last integral to the right is non-positive since $\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})$ is a positive measure. Hence

$$\int(-\psi)\mu\leq\int(-\psi)\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})+M\{\theta\}^{n}.$$

Iterating this argument yields

$$0\leq\int(-\psi)\mu\leq\int(-\psi)\theta^{n}+nM\{\theta\}^{n}.$$

Now $\int(-\psi)\theta^{n}$ is bounded above by some $C>0$ only depending on $\theta$, by compactness of $\{\psi\in\PSH(X,\theta)\mid\sup_{X}\psi=0\}$ and the fact that $\theta^{n}$ is an atomic measure supported at finitely many divisorial points. We conclude that

(3.3)

$$0\leq\int(-\psi)\mu\leq C+nM\{\theta\}^{n}$$

for some constant $C>0$ only depending on $\theta$, as long as $\psi$ is a bounded $\theta$-psh function with $\sup_{X}\psi=0$. If $\psi$ is now a possibly unbounded $\theta$-psh function normalized by $\sup_{X}\psi=0$, $\psi$ is the decreasing limit of the bounded $\theta$-psh functions $\psi_{m}:=\max\{\psi,-m\}$, so that (3.3) continues to hold, by monotone convergence. ∎