To ease notation, set $r^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $r:=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$. Let $\sigma$ be the smallest simplex of $\Delta({\mathcal{X}})$ containing $r(\sigma^{\prime})$. Write $Y$, $Y^{\prime}$ and $Y^{\prime\prime}$ for the strata of ${\mathcal{X}}_{0}$, ${\mathcal{X}}^{\prime}_{0}$ and ${\mathcal{X}}^{\prime\prime}_{0}$ corresponding to $\sigma$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$, respectively. The restrictions $r^{\prime}|_{\sigma^{\prime\prime}}\colon\sigma^{\prime\prime}\to\sigma^{\prime}$ and $r_{\sigma^{\prime}}\colon\sigma^{\prime}\to\sigma$ are given by ${\mathbb{Q}}$-linear maps, and we have induced morphisms $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$.

First suppose that $\sigma^{\prime\prime}$ is active for $r^{\prime}$ and $\sigma^{\prime}$ is active for $r$. Then $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are given by ${\mathbb{Q}}$-linear isomorphisms; hence so is the composition $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}$. Similarly, the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms; hence so is the composition $Y^{\prime\prime}\to Y$. It follows that $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$.

Conversely, suppose $\sigma^{\prime\prime}$ is active for $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$. Since the map $Y^{\prime\prime}\to Y$ is a bimeromorphic morphism, the map $Y^{\prime\prime}\to Y^{\prime}$ (resp. $Y^{\prime}\to Y$) must be injective (resp. surjective). In particular, $\dim Y^{\prime\prime}\leq\dim Y^{\prime}$ and $\dim Y\leq\dim Y^{\prime}$. Similarly, since the ${\mathbb{Q}}$-linear map defining $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}|_{\sigma^{\prime\prime}}=r|_{\sigma^{\prime}}\circ r^{\prime}|_{\sigma^{\prime\prime}}$ is an isomorphism, the ${\mathbb{Q}}$-linear map defining $r^{\prime}|_{\sigma^{\prime\prime}}$ (resp. $r|_{\sigma^{\prime}}$) must be injective (resp. surjective). In particular, $\dim\sigma^{\prime\prime}\leq\dim\sigma^{\prime}$ and $\dim\sigma\leq\dim\sigma^{\prime}$. Now

so we infer that $\dim Y^{\prime\prime}=\dim Y^{\prime}=\dim Y$ and $\dim\sigma=\dim\sigma^{\prime}=\dim\sigma^{\prime\prime}$. This further implies that the maps $Y^{\prime\prime}\to Y^{\prime}$ and $Y^{\prime}\to Y$ are bimeromorphic morphisms, and that the ${\mathbb{Q}}$-linear maps defining $r^{\prime}|_{\sigma^{\prime\prime}}$ and $r|_{\sigma^{\prime}}$ are isomorphisms. Hence $\sigma^{\prime\prime}$ and $\sigma^{\prime}$ are active for $r^{\prime}$ and $r$, respectively. ∎