Pick a closed point $\xi^{\prime\prime}\in\mathring{Y^{\prime\prime}}$ and set $\xi=\rho(\xi^{\prime\prime})\in\mathring{Y}$. We use the notation at the end of §6.2 with $p=d$. Namely, pick local coordinates $(z_{i})_{0\leq i\leq n}$ at $\xi$ and $(z^{\prime\prime}_{j})_{0\leq j\leq n}$ at $\xi^{\prime\prime}$ such that $E_{i}=\{z_{i}=0\}$ for $0\leq i\leq d$ and $E^{\prime\prime}_{j}=\{z_{j}^{\prime\prime}=0\}$ for $0\leq j\leq d$. We have $\rho^{*}z_{i}=u_{i}\prod_{j=0}^{d}(z^{\prime\prime}_{j})^{c_{ij}}$ for $0\leq i\leq d$, where $c_{ij}\in{\mathbb{Z}}_{\geq 0}$ and $u_{i}\in{\mathcal{O}}_{{\mathcal{X}}^{\prime\prime},\xi^{\prime\prime}}$ is a unit. Further, by Lemma 5.13, the matrix $(c_{ij})$ has determinant $\pm e_{\sigma}$, where $e_{\sigma}=m/\gcd(m,b_{\sigma})$.

Set

$$\Omega_{1}:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{d}}{z_{d}}{\quad\text{and}\quad}\Omega_{2}:=dz_{d+1}\wedge\dots\wedge dz_{n},$$

and define $\Omega_{1}^{\prime\prime}$, $\Omega^{\prime\prime}_{2}$ similarly. Then $\Omega:=\Omega_{1}\wedge\Omega_{2}$ and $\Omega^{\prime\prime}:=\Omega^{\prime\prime}_{1}\wedge\Omega_{2}^{\prime\prime}$ are local ${\mathbb{Q}}$-generators of $K^{\mathrm{log}}_{\mathcal{X}}$ and $K^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}}$ at $\xi$ and $\xi^{\prime\prime}$, respectively. Further,

$$\operatorname{Res}_{Y}(\Omega)=\Omega_{2}|_{Y}{\quad\text{and}\quad}\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime})=\Omega^{\prime\prime}_{2}|_{Y^{\prime\prime}}.$$

Now

$$\rho^{*}\Omega_{1}=\pm e_{\sigma}\Omega^{\prime\prime}_{1}+\frac{1}{z_{0}^{\prime\prime}\dots z^{\prime\prime}_{d}}\tilde{\Omega}^{\prime\prime}_{1},$$

where $\tilde{\Omega}^{\prime\prime}_{1}$ is a regular $(d+1)$-form vanishing at $\xi^{\prime\prime}$, and

$$\rho^{*}\Omega_{2}=q\Omega^{\prime\prime}_{2}+\tilde{\Omega}^{\prime\prime}_{2},$$

where $q\in{\mathcal{O}}_{{\mathcal{X}},\xi^{\prime\prime}}$ and $\tilde{\Omega}^{\prime\prime}_{2}$ is a regular $(n-d)$-form at $\xi^{\prime\prime}$ satisfying $\Omega^{\prime\prime}_{1}\wedge\tilde{\Omega}^{\prime\prime}_{2}=0$. On the one hand, this leads to

$$(\rho|_{Y^{\prime\prime}})^{*}\operatorname{Res}_{Y}(\Omega)=(\rho|_{Y^{\prime\prime}})^{*}(\Omega_{2}|_{Y})=q\Omega^{\prime\prime}_{2}|_{Y^{\prime\prime}}=q\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime}).$$

On the other hand, we also get

$$\rho^{*}\Omega=\pm qe_{\sigma}(1+h)\Omega^{\prime\prime},$$

with $q$ as above and $h\in{\mathcal{O}}_{{\mathcal{X}}^{\prime\prime},\xi^{\prime\prime}}$ vanishing along $Y^{\prime\prime}$.

Define $\Omega^{\mathrm{rel}}$ and $\Omega^{{}^{\prime\prime}\mathrm{rel}}$ by $\frac{d{t}}{{t}}\otimes\Omega^{\mathrm{rel}}=\Omega$ and $\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\Omega^{{}^{\prime\prime}\mathrm{rel}}=\Omega^{\prime\prime}$, respectively. Then

$$m\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\rho^{*}\Omega^{\mathrm{rel}}=\rho^{*}(\frac{d{t}}{{t}})\otimes\rho^{*}\Omega^{\mathrm{rel}}=\rho^{*}\Omega=\pm qe_{\sigma}(1+h)\Omega^{\prime\prime}=\pm qe_{\sigma}(1+h)\frac{d{t}^{\prime}}{{t}^{\prime}}\otimes\Omega^{{}^{\prime\prime}\mathrm{rel}},$$

so that

$$\rho^{*}\Omega^{\mathrm{rel}}=\pm\frac{e_{\sigma}}{m}q(1+h)\Omega^{{}^{\prime\prime}\mathrm{rel}}.$$

As a consequence,

$$\rho^{*}|{t}^{\kappa_{\min}}\Omega^{\mathrm{rel}}|_{\psi_{0}}=\frac{e_{\sigma}}{m}|q||(1+h)||({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}.$$

Since $h$ vanishes along $Y^{\prime\prime}$, this finally leads to

$$(\rho|_{Y^{\prime\prime}})^{*}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})=\frac{(\rho|_{Y^{\prime\prime}})^{*}|\operatorname{Res}_{Y}(\Omega)|^{2}}{(\rho|_{Y^{\prime\prime}})^{*}|{t}^{\kappa_{\min}}\Omega^{\mathrm{rel}}|^{2}_{\psi_{0}}}=\frac{|(\rho|_{Y^{\prime\prime}})^{*}(\Omega_{2}|_{Y})|^{2}}{\frac{e^{2}_{\sigma}}{m^{2}}|q|^{2}|({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}^{2}}\\ =\left(\frac{m}{e_{\sigma}}\right)^{2}\frac{|\operatorname{Res}_{Y^{\prime\prime}}(\Omega^{\prime\prime})|^{2}}{|({t}^{\prime})^{\kappa^{\prime}_{\min}}\Omega^{{}^{\prime\prime}\mathrm{rel}}|_{\psi^{\prime}_{0}}^{2}}=\left(\frac{m}{e_{\sigma}}\right)^{2}\operatorname{Res}_{Y^{\prime\prime}}({\mathcal{L}}^{{}^{\prime\prime}\#}),$$

which completes the proof since $e_{\sigma}=\frac{m}{\gcd(b_{\sigma},m)}$. ∎