We first notice that in the above statement, $U$ can be replaced by any smaller open set that meets all the log canonical centers: the closure of the restriction of $\sum^{j}_{i=\ell+1}\Delta_{i}|_{U}$ to $U\cap D$ will yield the same divisor on $D$.

Then by induction, we only need to treat the case where $D$ is a component of $\Delta$, say $\Delta_{1}$. If we take a log resolution $f:Y\to(Z,\Delta)$ and let $D^{\prime}=\Delta^{\prime}_{1}$ be the birational transform of $D$, then $\Delta_{D}$ can be computed as follows: if we write $f^{*}(K_{Z}+\Delta)|_{D^{\prime}}=K_{D^{\prime}}+\Delta_{D^{\prime}}$ then $\Delta_{D}=(f|_{D^{\prime}})_{*}(\Delta_{D^{\prime}})$. In particular, as $K_{Z}+\Delta$ is Cartier, we know that $\Delta_{D}$ is a integral divisor. Since it is effective, and all the components of $\Delta_{D}$ are log canonical centers of $(X,\Delta)$, we see that $\Delta_{D}$ must be equal to the closure of the restriction of $\sum^{j}_{i=2}\Delta_{i}|_{U}$. ∎

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

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