Let $\mathfrak{X}^{\prime}$ be a strongly nondegenerate strictly polystable formal scheme such that there is a surjective étale morphism $\varphi:\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$. Let $q\in\tilde{\mathfrak{X}}$ be the closed point corresponding to $\tau$. By Proposition 2.4 we have $\red_{\mathfrak{X}}^{-1}(q)=p_{\mathfrak{X}}^{-1}(\tau)$. Choose $q^{\prime}\in\tilde{\mathfrak{X}^{\prime}}$ with $\varphi(q^{\prime})=q$. By [Gub07, Proposition 2.9] we have that $\varphi$ induces an isomorphism $\red_{\mathfrak{X}^{\prime}}^{-1}(q^{\prime})\tilde{\rightarrow}p_{\mathfrak{X}}^{-1}(\tau)$. Hence the pullback of $\left(\mathcal{O}_{X^{\textup{an}}}\Big|_{p_{\mathfrak{X}}^{-1}(\tau)},\|\cdot\|\right)$ is the trivial bundle on $\red_{\mathfrak{X}^{\prime}}^{-1}(q^{\prime})$ endowed with the metric $\|1\|^{\prime}=e^{-h\circ p_{\mathfrak{X}}\circ\varphi}$. Since $p_{\mathfrak{X}}\circ\varphi=\varphi\circ p_{\mathfrak{X}^{\prime}}$ it follows that $\|\cdot\|^{\prime}$ is the metric associated to the function $h\circ\varphi$ on $\tau^{\prime}:=p_{\mathfrak{X}^{\prime}}(\red_{\mathfrak{X}^{\prime}}^{-1}(q^{\prime}))$ which is again rational piecewise affine linear by [Ber04, Theorem 6.1.1] and we may assume it is convex by definition. Let $y\in p_{\mathfrak{X}}^{-1}(\tau)$. In a neighbourhood of $p_{\mathfrak{X}^{\prime}}(y^{\prime})$ where $y^{\prime}\in p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})$ with $\varphi^{\textup{an}}(y^{\prime})=y$ we can write $h\circ\varphi^{\textup{an}}=\max_{i=1,...,s}h^{\prime}_{i}$ for suitable affine linear functions $h^{\prime}_{i}$ on $\tau^{\prime}$. Now as $\varphi^{\textup{an}}:p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})\rightarrow p_{\mathfrak{X}}^{-1}(\tau)$ is an isomorphism we have $h=\max_{i=1,...,s}h_{i}$ where $h_{i}$ are the piecewise affine linear functions on $\tau$ satisfying $h^{\prime}_{i}=h_{i}\circ\varphi^{\textup{an}}$. Now the metrics associated to the $h^{\prime}_{i}$ are piecewise $\mathbb{Q}$-linear and semipositive in $y^{\prime}$ by the same argument as in the proof of Proposition 5.6. Hence the piecewise $\mathbb{Q}$-linear metrics associated to the $h_{i}$ extend from a compact strictly $K$-analytic neighbourhood of $y$ to global metrics by [GM19, Proposition 2.7] which are semipositive in $y$. Now as $\|\cdot\|$ is locally around $y$ given as the minimum of these metrics, also $\|\cdot\|$ is a piecewise $\mathbb{Q}$-linear metric which is semipositive in $y$ by Proposition 3.11. ∎