Let $y\in p_{\mathfrak{X}}^{-1}(\tau)$ and $x:=p_{\mathfrak{X}}(y)\in\tau$. There is an open neighbourhood $U$ of $x$ in $\tau$ such that we can write $h\Big|_{U}=\max_{i=1,...,s}h_{i}\Big|_{U}$ for suitable rational affine linear functions $h_{i}$ on $\tau$. After passing to some multiple, each $h_{i}$ induces a formal metric on $\mathfrak{X}^{\prime an}$ by Proposition 2.11 where $\mathfrak{X}^{\prime}$ is defined as in Remark 5.1. Therefore the $h_{i}$ induce piecewise $\mathbb{Q}$-linear metrics on $\mathcal{O}_{\mathfrak{X}^{\textup{an}}}\Big|_{p_{\mathfrak{X}}^{-1}(\tau)}$ since $p_{\mathfrak{X}}^{-1}(\tau)\subseteq\mathfrak{X}^{\prime an}$. Hence in the neighbourhood $p_{\mathfrak{X}}^{-1}(U)$ of $y$, the metric induced by $h$ is given as the minimum of the metrics corresponding to the $h_{i}$, which are semipositive at $y$ by Lemma 2.13. Indeed let $(\mathfrak{X}^{\prime\prime}_{i},\mathfrak{L}_{i})$ be a formal model of the trivial bundle associated to $h_{i}$ as obtained by Proposition 2.11. Then by [GK19, Proposition 6.5] (the proof of the implication we need does neither use that $K$ is algebraically closed nor that the generic fibre is algebraic) it is enough to show that $\Deg_{\mathfrak{L}_{i}}(Y)\geq 0$ for any closed curve $Y$ in $\tilde{\mathfrak{X}}^{\prime\prime}_{i}$ with $Y\subseteq\red(p_{\mathfrak{X}^{\prime\prime}_{i}}^{-1}(\tau))$ but by Lemma 2.13 we even have equality. Now we extend the metrics induced by the $h_{i}$ from a compact strictly $K$-analytic neighbourhood of $y$ to $X^{\textup{an}}$ by [GM19, Proposition 2.7] and then it follows from Proposition 3.11 that $||\cdot||$ is semipositive at $y$. ∎