Theorem 7.1 fails in general when $X$ does not have semistable reduction. Indeed, the semistable reduction theorem [KKMS] shows that the base change $p\colon X^{\prime}\to X$ to ${\mathbb{C}}(\!({t}^{1/m})\!)$ has semistable reduction for some $m$ divisible enough. By Lemma 5.14, $\dim\operatorname{Sk}(X^{\prime})=n$, and $\mu_{\log|\eta^{\prime}|^{\#}}$ is thus a multiple of the integral Lebesgue measure $\lambda^{\prime}$ of $\operatorname{Sk}(X^{\prime})$, by Theorem 7.1. By Theorem 6.7, $\mu_{\log|\eta|^{\#}}=m^{-n}p_{*}\lambda^{\prime}$. However, $p_{*}\lambda^{\prime}$ is not proportional to the integral Lebesgue measure $\lambda$ of $\operatorname{Sk}(X)$ in general. Indeed, for each $n$-simplex $\sigma$ of $\Delta({\mathcal{L}})$, Lemma 5.13 shows that $(p_{*}\lambda^{\prime})_{\sigma}=m^{n}b_{\sigma}\lambda_{\sigma}$, and $b_{\sigma}$ is in general not independent of $\sigma$.