By Proposition 2.9(vi) and the continuity of the Monge–Ampère measure given in [BFJ15, Thm. 3.1], we may assume that $\theta\in\mathcal{Z}^{1,1}(X)_{\mathbb{Q}}$. Using Proposition 2.9(vii), we may assume that $\theta$ is induced by a line bundle of a model of $X$. Then the claim follows from [BGJKM16, Thm. 6.3.2] as ${P}_{\theta}(f)$ is continuous by Theorem 8.2. ∎