By running MMP for $\mathscr{Y}$ over $\mathscr{C}$, we know that $X$ has a good minimal dlt model $\mathscr{X}$ with reduced special fiber (see [Fu11] or [HX13]). Then one sees as in the proof of Theorem 4.1.4 that $K_{\mathscr{X}}$ is trivial over $\mathscr{C}$. Our assumption on the dimension of $\mathrm{Sk}(X_{K})$ implies that the minimal log canonical centers of $(\mathscr{X},\mathscr{X}_{s})$ are points. Let $D$ be a one-dimensional log canonical center, and let $D_{i}$ ($1\leq i\leq\ell$) be the 0-dimensional log canonical centers contained in $D$. From Lemma 4.1.6, we know that

$$(K_{\mathscr{X}}+\mathscr{X}_{s})|_{D}=K_{D}+\sum^{\ell}_{i=1}D_{i}\sim 0.$$

Thus $D$ is a rational curve and $\ell=2$, which means that $\mathrm{Sk}(X_{K})$ is closed. ∎