(1) The condition that $K_{X}$ is semi-ample over $C$ is obviously necessary, since for every $dlt$-model $\mathscr{X}$ of $X$, the divisor $K_{X}$ is $\mathbb{Q}$-linearly equivalent to the restriction of $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ to $X$. Conversely, assume that $K_{X}$ is semi-ample over $C$, and let $\mathscr{Y}$ be a proper $snc$-model of $X$. Then applying [HX13, 2.12] to the $dlt$-pair $(\mathscr{Y},(\mathscr{Y}_{s})_{\mathrm{red}})$, we see that $X$ has a good minimal $dlt$-model. Condition (1) of [HX13, 2.12] follows from our assumption, and condition (2) follows from the following observation. Let $m$ be a positive integer such that $m(K_{\mathscr{Y}}+(\mathscr{Y}_{s})_{\mathrm{red}})$ is Cartier. Over a sufficiently small open neighbourhood of $s$ in $\mathscr{C}$, we have an isomorphism of $\mathcal{O}_{\mathscr{C}}$-algebras

$$R(\mathscr{Y}/\mathscr{C},m(K_{\mathscr{Y}}+(\mathscr{Y}_{s})_{\mathrm{red}}))\cong R(\mathscr{Y}/\mathscr{C},m(K_{\mathscr{Y}}+(\mathscr{Y}_{s})_{\mathrm{red}})-\mathscr{Y}_{s}),$$

where $R(\mathscr{Y}/\mathscr{C},L):=\bigoplus_{j\geq 0}\pi_{*}(\mathcal{O}_{\mathscr{Y}}(jL))$ with $\pi:\mathscr{Y}\to\mathscr{C}$ the structural morphism. Thus it suffices to show that

$$\mathcal{A}=R(\mathscr{Y}/\mathscr{C},m(K_{\mathscr{Y}}+(\mathscr{Y}_{s})_{\mathrm{red}})-\mathscr{Y}_{s})$$

is a finitely generated $\mathcal{O}_{\mathscr{C}}$-algebra. If we denote by $M$ the maximum of the multiplicities of the components in $\mathscr{Y}_{s}$ then we may assume that $m>M$, so that

$$(\mathscr{Y},(\mathscr{Y}_{s})_{\mathrm{red}}-\frac{1}{m}\mathscr{Y}_{s})$$

is a $klt$ pair. Hence, the finite generation of $\mathcal{A}$ follows from [BCHM10].

(2) It is already observed in Definition 15 of [dFKX12] that this follows from the proof of [KM98, 3.52]. ∎