Applying [MN13, 3.1.7] to the proper morphism $h^{-1}(\mathscr{X}^{\mathrm{snc}})\to\mathscr{X}^{\mathrm{snc}}$, we see that $\mathrm{Sk}(\mathscr{X})$ is contained in $\mathrm{Sk}(\mathscr{Y})$. Moreover, it follows from Lemma 3.2.3 that for every point $x$ of $S$, the reduction $\mathrm{red}_{\mathscr{X}}(x)$ must be contained in $\mathscr{X}^{\mathrm{snc}}$. Now let $x$ be any point in $\mathrm{Sk}(\mathscr{Y})$ such that $\mathrm{red}_{\mathscr{X}}(x)$ lies in $\mathscr{X}^{\mathrm{snc}}$. We must show that $v_{x}(\Delta)=v_{x}((\mathscr{Y}_{s})_{\mathrm{red}})$ if and only if $x$ lies in $\mathrm{Sk}(\mathscr{X})$, or, equivalently, $x$ is equal to its projection

$$x^{\prime}=\rho_{\mathscr{X}}(x)$$

to the skeleton of $\mathscr{X}$. Let $\omega$ be a local generator of $\omega_{\mathscr{X}^{\mathrm{snc}}/\mathscr{C}}$ at $\mathrm{red}_{\mathscr{X}}(x)$. It induces a rational section of the canonical bundle $\omega_{X_{K}/K}$ by base change. By [MN13, 4.4.5], we know that $x=x^{\prime}$ if and only if

$$\mathrm{wt}_{\omega}(x)=\mathrm{wt}_{\omega}(x^{\prime}).$$

Since the divisor of $\omega$ is zero in a neighbourhood of $\mathrm{red}_{\mathscr{X}}(x^{\prime})$, we have

$$\mathrm{wt}_{\omega}(x^{\prime})=v_{x^{\prime}}((\mathscr{X}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}}).$$

On the other hand, computing $\mathrm{wt}_{\omega}(x)$ on the model $\mathscr{Y}$ we get

$$\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+(\mathscr{Y}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}})+v_{x}((\mathscr{Y}_{s})_{\mathrm{red}}-\Delta).$$

Thus we see that $\mathrm{Sk}(\mathscr{X})=S$. ∎