For notational simplicity, we do the proof for $n=3$, the $n=2$ case being even simpler. By $\mathfrak{S}_{5}$-symmetry, it is enough to check this on the open $U_{0}$.

On $U_{0}$ the $\mathbb{Z}$-affine structure induced by $\pi$ matches the one associated with a minimal model $\mathscr{X}^{\prime}$, such the strict transform of $D_{0}$ inside $\mathscr{X}^{\prime}$ is isomorphic to $\mathbb{C}\mathbb{P}^{3}$ and the hypotheses of Theorem B hold for the stratum $D_{0}$. For the affine structure induced by an affinoid torus fibration, $\mathbb{Z}$-affine functions on $U_{0}$ are given by $-\log\lvert h\rvert$, where $h$ is a non-vanishing analytic function on $\pi^{-1}(U_{0})$ (see Section 1.6), and $\pi^{-1}(U_{0})$ is the generic fiber (in the sense of Berkovich) of $\widehat{\mathscr{X}^{\prime}_{/D_{0}}}$ (see Section 1.5).

Using the results of Section 2, we may assume that we are working on the generic fiber of $\widehat{\mathscr{N}_{/D_{0}}}$, which we denote by $\mathfrak{N}_{D_{0}}$; this is an open subset of the analytification of the torus $\mathbb{T}$ of $\mathscr{N}=\mathcal{N}\times_{\mathbb{A}^{1}_{k}}R$, where $\mathcal{N}=\nu_{D_{0}/\mathscr{X}^{\prime}}$. Thus we replace $\pi:\pi^{-1}(U_{0})\rightarrow U_{0}$ with $\val:\mathfrak{N}_{D_{0}}\subset\mathbb{T}^{\an}\rightarrow\Star(e_{0})\simeq U_{0}$.

The torus $\mathbb{T}_{\mathcal{N}}$ of $\mathcal{N}$ is the direct product of the torus of $D_{0}$ with $\mathbb{G}_{m,k}$, i.e. in coordinates

$$\mathbb{T}_{\mathcal{N}}=\mathbb{T}_{D_{0}}\times_{k}\mathbb{G}_{m,k}=\Spec\,k\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big].$$

The normal bundle is endowed with a morphism $t:\mathcal{N}\rightarrow\mathbb{A}^{1}_{k}$, whose restriction $\mathbb{T}_{\mathcal{N}}\rightarrow\mathbb{G}_{m,k}$ corresponds to the morphism of rings

$$k[t^{\pm}]\rightarrow k\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big],\quad t\mapsto u\prod_{i=1}^{3}\frac{z_{i}}{z_{4}}.$$

We obtain that

$$\mathbb{T}=\mathbb{T}_{\mathcal{N}}\times_{\mathbb{G}_{m,k}}K=\Spec\frac{K\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm},u^{\pm}\Big]}{t-u\prod_{i=1}^{3}\frac{z_{i}}{z_{4}}}=\Spec\,K\Big[\Big(\frac{z_{1}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{2}}{z_{4}}\Big)^{\pm},\Big(\frac{z_{3}}{z_{4}}\Big)^{\pm}\Big],$$

so that $\mathbb{Z}$-affine functions on $\Star(e_{0})$ are integral linear combinations of the $-\log\lvert\frac{z_{i}}{z_{4}}\rvert$, for $i=1,2,3$. But those functions are precisely $m_{1}-m_{4}$, $m_{2}-m_{4}$, $m_{3}-m_{4}$, i.e. $m\in M$ satisfying $\langle m,e_{0}\rangle=0$ and generating the $\mathbb{Z}$-linear functions on $U_{0}$ in [Li19]. ∎