By Lemma B.3 we may approximate $\varphi$ by functions of the form $\frac{1}{d_{m}}\log|\mathfrak{a}_{m}|$ for some vertical coherent fractional ideals $\mathfrak{a}_{m}$ on $\mathscr{X}$. On the generic fibre of a building block $\mathfrak{U}$, the function $\log|\mathfrak{a}_{m}|$ is given as the maximum of the functions $\log|f|$ where $f$ runs through a finite set of generators of $\mathfrak{a}_{m}\Big|_{\mathfrak{U}}$. Since the properties we are looking for are stable under taking the maximum, these functions have them by Lemma B.1. But they are also stable under uniform limits so we are done. ∎