As we mentioned above, this result is essentially contained in [KK10, Ko11]. Using the terminology there, properties (1)-(3) of a pseudo-manifold all follow from the fact that two minimal log canonical centers of a log crepant structure are $\mathbb{P}^{1}$-linked in the sense of Definition 9 in [Ko11]. We will now explain this in more detail. We denote by $n$ the relative dimension of $X$ over $C$.

By Theorem 2.2.6(1), there exists a a good minimal $dlt$-model $\mathscr{X}$ of $X$ over $\mathscr{C}$. By Theorem 3.3.4, we have $\mathrm{Sk}(X_{K})=\mathrm{Sk}(\mathscr{X})$. As a triangulation on $\mathrm{Sk}(\mathscr{X})$, we take the first barycentric subdivision of the simplicial structure on $\mathrm{Sk}(\mathscr{X})$. This barycentric subdivision is necessary to guarantee that the intersection of two faces is a codimension one face of both, rather than a union of faces (think of a type $I_{2}$ degeneration of elliptic curves, whose skeleton consists of two vertices joined by two edges).

We choose an integer $m>0$ such that $mK_{X}\sim 0$. Since the divisor $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$ and trivial over $C$, we see that $mK_{\mathscr{X}}+m(\mathscr{X}_{s})_{\mathrm{red}}$ must be a multiple of $\mathscr{X}_{s}$ and thus trivial over $\mathscr{C}$. Thus we can apply Theorem 10 in [Ko11] to the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ over $\mathscr{C}$. It states that every two minimal log canonical centers $D$ and $D^{*}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are $\mathbb{P}^{1}$-linked. This means, in particular, that they have the same dimension, say $n-d$, and that there exist a sequence of $(n-d+1)$-dimensional log canonical centers $E_{1},E_{2},\ldots,E_{\ell}$ and a sequence of $(n-d)$-dimensional log canonical centers $D=D_{0},D_{1},\ldots,D_{\ell}=D^{*}$ such that $D_{i-1},D_{i}\subset E_{i}$ for $1\leq i\leq\ell$. In this way, we obtain properties (1) and (3) of a pseudo-manifold with boundary.

If we have two minimal log canonical centers $D_{1},\,D_{2}$ of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$, contained in an $(n-d+1)$-dimensional log canonical center $E$, and if we write

for some $\Delta\geq 0$, then $(E,D_{1}+D_{2}+\Delta)$ is again a $dlt$-pair [Ko13, 4.19]. Moreover, $D_{1}$ cannot intersect $D_{2}$ or $\lfloor\Delta\rfloor$ because the intersection would be a union of log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$, which contradicts the minimality of $D_{1}$. Thus we are in the situation of the second part of the proof of Theorem 10 in [Ko11]. That proof shows that $D_{1}$ and $D_{2}$ are the only log canonical centers of $(E,D_{1}+D_{2}+\Delta)$. Property (2) follows. ∎