We may assume that $K^{\prime}$ is Galois over $K$. Let $d$ be the ramification index of $K^{\prime}$ over $K$. We will prove that

for every divisorial point $y$ on $(Y^{\prime})^{\mathrm{an}}$ (see [MN13, 2.4.10] for the notion of divisorial point). This immediately implies the statement in the lemma, since $\mathrm{Sk}(X,\omega)$ is the closure of the set of divisorial points where the weight function reaches its minimal value [MN13, 4.5.1].

We denote by $R^{\prime}$ the integral closure of $R$ in $K^{\prime}$. Let $\mathscr{Y}^{\prime}$ be a regular separated $R^{\prime}$-scheme of finite type with irreducible special fiber $\mathscr{Y}^{\prime}_{k}$, endowed with an isomorphism of $K^{\prime}$-schemes $\mathscr{Y}^{\prime}_{K^{\prime}}\to Y^{\prime}$. Let $y$ be the unique point in $\mathrm{red}_{\mathscr{Y}^{\prime}}^{-1}(\xi)$, where $\xi$ denotes the generic point of $\mathscr{Y}^{\prime}_{k}$. Removing a closed subset of $\mathscr{Y}^{\prime}_{k}$ if necessary, we can find a regular separated $R$-scheme of finite type $\mathscr{Y}$ and an isomorphism $\mathscr{Y}_{K}\to Y$ such that $\mathscr{Y}^{\prime}$ is an open subscheme of the normalization of $\mathscr{Y}\times_{R}R^{\prime}$. Then $\mathrm{red}_{\mathscr{Y}}(\pi(y))$ is a generic point of $\mathscr{Y}_{k}$.

If we use the notations from (3.2) and denote by $(S^{\prime})^{+}$ the log scheme associated to $R^{\prime}\setminus\{0\}\to R^{\prime}$, then the $(S^{\prime})^{+}$-log scheme $(\mathscr{Y}^{\prime})^{+}$ is isomorphic to an open log subscheme of the $fs$ base change of $\mathscr{Y}^{+}$ from $S^{+}$ to $(S^{\prime})^{+}$. Since log differentials are compatible with $fs$ base change, we can deduce from the description of the weight function in (3.2) that

(the scaling factor $d$ is caused by the renormalization of the discrete valuation on $K^{\prime}$). ∎