Reducing $f$ modulo $t$, we obtain an isomorphism of $k$-schemes $\mathscr{Y}_{k}\to\mathscr{Z}_{k}$ and, by taking the dual intersection complexes, an isomorphism of simplicial spaces with piecewise $\mathbb{Z}$-affine structure $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$. We will prove that this isomorphism maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$.

We use the notations from Proposition 4.2.2(2) and we set $s^{+}=s_{1}^{+}$. We denote by $\mathscr{Y}^{+}_{k}$ the log scheme $\mathscr{Y}^{+}\times_{S^{+}}s^{+}$ obtained by restricting the log structure on $\mathscr{Y}^{+}$ to the special fiber $\mathscr{Y}_{k}$ of $\mathscr{Y}$. It follows from [IKN05, 7.1] that

$$\Omega:=H^{0}(\mathscr{Y},\omega_{\mathscr{Y}^{+}/S^{+}})$$

is a free $R$-module of rank one and that the reduction map

$$\Omega\otimes_{R}k\to\Omega_{k}:=H^{0}(\mathscr{Y}_{k},\omega_{\mathscr{Y}^{+}_{k}/s^{+}})$$

is an isomorphism. Let $\omega$ be a generator of the $R$-module $\Omega$ and denote by $\omega_{k}$ its image in $\Omega_{k}$. By (3.2), the generic point $\xi$ of an irreducible component $E$ of $\mathscr{Y}_{k}$ is $\omega$-essential in the sense of [MN13, 4.5.4] if and only if $\omega_{k}$ generates $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at the point $\xi$. Moreover, the skeleton $\mathrm{Sk}(\mathscr{Y}_{K})=\mathrm{Sk}(\mathscr{Y}_{K},\omega)$ is the simplicial subspace of $\mathrm{Sk}(\mathscr{Y})$ spanned by the vertices corresponding to such points $\xi$ [MN13, 4.5.5]. However, for every integer $d>0$, the stalk of $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at $\xi$ is generated by global sections if and only if $\omega_{\mathscr{Y}^{+}\times_{S^{+}}s_{d}^{+}/s_{d}^{+}}$ is generated by global sections at any point lying above $\xi$, by the base change property in [IKN05, 7.1]. The analogous statements hold for $\mathscr{Z}$. Thus it follows from Proposition 4.2.2(2) that the isomorphism $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$ maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$. ∎