In the $G$-topology on ${X^{{\mathrm{an}}}}$ induced by the strictly $K$-affinoid domains, a $\mathbb{Z}$-model function is given locally by $-\log|\gamma|$ for a rational function $\gamma$ on $X$. Hence (a) follows from [GRW16, Prop. 5.6]. Property (b) was proven in [GH15, Prop. B.7] for any dimension.

To prove (c), we choose a subdivision of $S(\mathscr{X})$ as in Definition 3.4 for $G$. As in [BPR13, §3], this subdivison is the skeleton of a strictly semistable model $\mathscr{X}^{\prime}$ dominating $\mathscr{X}$ and with the same retraction $\tau$. Then (c) follows from (b). ∎