Proposition 19.4.32CFE94018A80A

Proof. Thanks to proposition 16.6.10, we may assume that $\varphi $ is $\mathcal{I}$-model. By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ with analytic singularities in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ j\xrightarrow {d_ S}\varphi $. It is clear that $\pi ^*\varphi _ j\xrightarrow {d_ S}\pi ^*\varphi $. By theorem 19.4.30, we may then reduce to the case where $\varphi $ has analytic singularities. In this case, it suffices to apply 19.4.23. $\square$