Pick a representative ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ of $\psi^{\#}$ such that ${\mathcal{L}}$ is defined on a proper snc model ${\mathcal{X}}$. Let ${\mathcal{X}}^{\prime}$ be the normalized base change by ${t}={t}^{\prime m}$.

Let $\sigma$ be a $d$-dimensional face of $\Delta({\mathcal{L}})$. By Lemma 5.13, $p^{-1}(\sigma)$ is the union of $g_{\sigma}$ distinct isomorphic faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}})$ such that

Further, the induced map $Y^{\prime}_{\sigma^{\prime}_{\alpha}}\to Y$ is generically finite, of degree $f_{\sigma}$ independent of $\alpha$, and we have $f_{\sigma}g_{\sigma}=\gcd(m,b_{\sigma})$. Pick a toroidal modification ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}^{\prime}$ with ${\mathcal{X}}^{\prime\prime}$ snc, denote by $\rho\colon{\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ the composition, and set ${\mathcal{L}}^{\prime\prime}:=\rho^{*}{\mathcal{L}}$.

Each face $\sigma^{\prime}_{\alpha}$ above is subdivided into simplices $\sigma^{\prime\prime}_{\alpha\beta}$ of $\Delta({\mathcal{L}}^{\prime\prime})$ of dimension $d$, each corresponding to a stratum $Y^{\prime\prime}_{\alpha\beta}$ of ${\mathcal{X}}^{\prime\prime}_{0}$, and $\rho|_{Y^{\prime\prime}_{\alpha\beta}}\colon Y^{\prime\prime}_{\alpha\beta}\to Y$ is generically finite, of degree $f_{\sigma}$. Further, (6.2) implies that

We shall need the following result:

Grant this result for the moment. Lemma 6.8 implies

and hence

thanks to (6.3) and (6.4). ∎