We shall reduce the statement to a geometric assertion that can be proved using asymptotic multiplier ideals.

First, we may assume that $u\in\mathcal{D}(X)$, thanks to Corollary 2.3 and (vi) of Proposition 8.2.

Second, we can reduce to the case when $\theta\in N^{1}(\mathcal{X}/S)_{\mathbf{Q}}$ is a rational class, using (vii) of Proposition 8.2.

Third, we may further reduce to the case $u=0$, after replacing $\theta$ with $\theta+dd^{c}u$, using (v) of Proposition 8.2.

After scaling, we may finally assume that $\theta$ is the curvature form of a model metric determined by a line bunle $\mathcal{L}$ on some model $\mathcal{X}$. Now we conclude the proof using the following result. ∎