We do not know whether the function $\varphi_{x}$ in (8.5) is necessarily a model function. This is the case on a toric variety, see Proposition 9.1 below, but we suspect the answer is no in general.

Pick an SNC model $\mathcal{X}$, an extension $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ of $L$, let $\omega$ be the curvature form of the model metric defined by $\mathcal{L}$. Let also $E$ be a component of $\mathcal{X}$ corresponding to the divisorial point $x=x_{E}$. We have $\varphi_{x}=P_{\omega}(-f_{E})$ up to a constant. On the other hand, by [BFJ11, Theorem 8.5],

where $\mathfrak{a}_{m}$ denotes the base-ideal of $m\mathcal{L}^{\prime}$ with $\mathcal{L}^{\prime}:=\mathcal{L}-E$. As a consequence, $\varphi_{x}$ is indeed a model function as soon as the graded $S$-algebra $\bigoplus_{m\geq 0}H^{0}(\mathcal{X},m\mathcal{L}^{\prime})$ is finitely generated. Building on Nakayama’s counterexample to the existence of Zariski decompositions [Nak04], it is reasonable to expect this algebra not to be finitely generated in general, and to subsequently prove that $\varphi_{x}$ is not a model function.