Pick any regular model $\mathcal{X}$. Then the linear map $N^{1}(\mathcal{X}/S)\to N^{1}(X)$ is surjective hence open. It is thus enough to prove the following claim: let $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ have ample image in $N^{1}(X)$, and assume that $\theta_{\mathcal{X}}$ is the limit of a sequence $\theta_{m,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. If the corresponding forms $\theta_{m}\in\mathcal{Z}^{1,1}(X)$ all satisfy the orthogonality property, then so does $\theta$.

Let $f\in C^{0}(X)$. By Proposition 2.15 we have $P_{\theta_{m}}(f)\to P_{\theta}(f)$ uniformly on $X$. We claim that

$$(\theta_{m}+dd^{c}P_{\theta_{m}}(f))^{n}\to(\theta+dd^{c}P_{\theta}(f))^{n}$$

with uniformly bounded mass. Since $(P_{\theta_{m}}(f)-f)\to(P_{\theta}(f)-f)$ uniformly on $X$, we have as before

$$\int\left(P_{\theta}(f)-f\right)\left(\theta+dd^{c}\ P_{\theta}(f)\right)^{n}=\lim_{m}\int\left(P_{\theta_{m}}(f)-f\right)\left(\theta+dd^{c}\ P_{\theta_{m}}(f)\right)^{n}=0$$

which concludes the proof.

To prove the claim, pick any model function $g\in\mathcal{D}(X)$, and fix $\varepsilon>0$. By Corollary 2.12, we can find a $\theta$-psh model function $\varphi$ such that $\sup|\varphi-P_{\theta}(f)|\leq\varepsilon$. We then have

$$I_{m}:=\left|\int g\,(\theta_{m}+dd^{c}P_{\theta_{m}}(f))^{n}-\int g\,(\theta+dd^{c}P_{\theta}(f))^{n}\right|\leq\\ \left|\int g\,(\theta_{m}+dd^{c}P_{\theta_{m}}(f))^{n}-\int g\,(\theta_{m}+dd^{c}\varphi)^{n}\right|+\left|\int g\,(\theta_{m}+dd^{c}\varphi)^{n}-\int g\,(\theta+dd^{c}\varphi)^{n}\right|+\\ \left|\int g\,(\theta+dd^{c}\varphi)^{n}-\int g\,(\theta+dd^{c}P_{\theta}(f))^{n}\right|$$

Using integration by parts, the last term can be bounded as follows.

$$\left|\int g\,(\theta+dd^{c}\varphi)^{n}-\int g\,(\theta+dd^{c}P_{\theta}(f))^{n}\right|=\\ \left|\int(\varphi-P_{\theta}(f))\,dd^{c}g\wedge\sum_{i=0}^{n-1}(\theta+dd^{c}\varphi)^{i}\wedge(\theta+dd^{c}P_{\theta}(f))^{n-i-1}\right|\leq C\varepsilon$$

where $\omega$ is a fixed form such that $(\omega+dd^{c}g)$ is semipositive, and $C=2\{\omega\}\,\{\theta\}^{n-1}$. In a similar way, the first term is bounded from above by

$$\left|\int g\,(\theta_{m}+dd^{c}P_{\theta_{m}}(f))^{n}-\int g\,(\theta_{m}+dd^{c}\varphi)^{n}\right|\leq C\,\sup|P_{\theta_{m}}(f)-\varphi|\leq 2C\varepsilon~,$$

for $m$ large enough. Finally $g$ and $\varphi$ being model functions, the second term tends to zero as $m\to\infty$, and we get $\limsup_{m}I_{m}\leq 3C\varepsilon$. We conclude by letting $\varepsilon\to 0$. ∎