Set $F:=\mu^{*}\mu_{*}G-G\in\Div_{0}(\mathcal{Y})_{\mathbf{Q}}$. The divisor $G$ is $\mu$-nef since $\mu^{*}(\rho^{*}\theta_{\mathcal{X}})+G$ is nef by assumption, and Lemma 1.6 therefore implies that $F$ is effective.

Let $W$ be the closure of the center of $v$ on $\mathcal{Y}$. Since the center of $v$ on $\mathcal{X}^{\prime}$ is the generic point of $E^{\prime}_{J}$, we must have $\mu(W)=E^{\prime}_{J}$. Note, however, that we do not claim $\dim W=\dim E^{\prime}_{J}$.

By Theorem 3.11, the function $\varphi_{\mu_{*}G}$ is affine on the face $\sigma^{\prime}_{J}$ of $\Delta^{\prime}$. But $\varphi_{G}$ is also affine on $\sigma^{\prime}_{J}$ by assumption, and we have

$$\varphi_{G}(e^{\prime}_{j})=b^{\prime}_{j}\ord_{E^{\prime}_{j}}(G)=\varphi_{\mu_{*}G}(e^{\prime}_{j})\quad\text{for all $j\in J$}.$$

It follows that $\varphi_{F}\equiv 0$ on $\sigma^{\prime}_{J}$, and in particular $v(F)=0$. But this means precisely that $W$ is not contained in $\supp F$, so that $F|_{W}$ is an effective $\mathbf{Q}$-Cartier divisor. Hence

$$\mu^{*}(\rho^{*}\theta_{\mathcal{X}}+\mu_{*}G)|_{E^{\prime}_{J}}=(\pi^{*}\theta_{\mathcal{X}}+\mu^{*}\mu_{*}G)|_{W}=(\pi^{*}\theta_{\mathcal{X}}+G)|_{W}+F|_{W}$$

is the sum of a nef class and an effective class. We conclude by Lemma 6.6 below. ∎