By the definition of a $dlt$-model, we know that $\Delta\leq(\mathscr{Y}_{s})_{\mathrm{red}}$. Thus it suffices to show that these divisors are different locally at $\mathrm{red}_{\mathscr{Y}}(x)$. Since $x$ lies on $\mathrm{Sk}(\mathscr{Y})$, its reduction $\mathrm{red}_{\mathscr{Y}}(x)$ is a generic point of the intersection of the irreducible components of $\mathscr{Y}_{s}$ that contain $\mathrm{red}_{\mathscr{Y}}(x)$. Thus if we denote by $h^{\prime}:\mathscr{Y}^{\prime}\to\mathscr{Y}$ the blow-up of $\mathscr{Y}$ at the closure of $\mathrm{red}_{\mathscr{Y}}(x)$, then $\mathscr{Y}^{\prime}$ is again an $snc$-model of $X$.

We denote by $\Delta^{\prime}$ the log pullback of $\Delta$ to $\mathscr{Y}^{\prime}$. The image of the exceptional divisor $E$ of $h^{\prime}$ in $\mathscr{X}$ is the closure of $\mathrm{red}_{\mathscr{X}}(x)=h(\mathrm{red}_{\mathscr{Y}}(x))$ and thus disjoint from $\mathscr{X}^{\mathrm{snc}}$. By the definition of a $dlt$-model, we know that the multiplicity of $E$ in $\Delta^{\prime}$ is strictly smaller than $1$. Since the log pullback of $(\mathscr{Y}_{s})_{\mathrm{red}}$ to $\mathscr{Y}^{\prime}$ is equal to $(\mathscr{Y}^{\prime}_{s})_{\mathrm{red}}$, we see that $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally at $\mathrm{red}_{\mathscr{Y}}(x)$. ∎