Over $\textrm{Int}(\tau)$ of any $2$-dimensional face $\tau\subset\Sk(\mathscr{X})$, $\overline{\rho}$ is equal to $\rho_{\mathscr{Z}}$, hence is an affinoid torus fibration. Around any vertex $v_{D}$, $\overline{\rho}$ is equal to $\rho_{\mathscr{X}_{ijk}}$ for any triple such that $D\neq D_{i},D_{j},D_{k}$. It follows from Section 3.3.1 that $\overline{\rho}$ is an affinoid torus fibration around $v_{D}$. We denote by $p_{0}+p_{\infty}$ the boundary of $C_{e}$, with $p_{0}=C_{e}\cap D_{i_{0}}$ and $p_{\infty}=C_{e}\cap D_{i_{\infty}}$; we write $a_{e,0}=v_{D_{e_{1}}}$ and $a_{e,5}=v_{D_{e_{2}}}$, and denote by $(\cdot,\cdot)$ the open segment joining two points. Then, for $j\in\{0,\ldots,4\}$, $\overline{\rho}$ is equal to $\rho_{\mathscr{X}_{e,j}}$ over $\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup(a_{e,j},a_{e,j+1})$, thus is an affinoid torus fibration. We conclude that $\overline{\rho}$ is an affinoid torus fibration away from the points $a_{e,i}$ for $i\in\{1,\ldots,4\}$ .

For a singular point $a_{e,i}$, we consider a loop $\gamma$ around it and contained in $\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup(a_{e,i-1},a_{e,i})\cup(a_{e,i},a_{e,i+1})$. We apply Corollary 3.2.4 to compute the monodromy along $\gamma$: the numbers $b_{e_{1},\mathscr{X}_{e,i-1}}$ and $b_{e_{1},\mathscr{X}_{e,i}}$ differ by $1$, as the model $\mathscr{X}_{e,i}$ has an additional exceptional curves in $D_{e_{1}}$ with respect to $\mathscr{X}_{e,i-1}$. Therefore, we obtain

with respect to the basis $(v_{D_{i_{0}}},v_{D_{e_{1}}})$ and origin $v_{D_{e_{2}}}$. ∎