Let $\mathscr{X}^{\prime}$ be a projective snc model dominating $\mathscr{X}$ and write $h:\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$. We denote by $D_{0},\ldots,D_{p}$ the components of $\mathscr{X}_{0}$ containing $Y$.

Let $Z^{\prime}\subseteq D^{\prime}_{0}\cap\ldots\cap D^{\prime}_{q}$ be any stratum of $\mathscr{X}_{0}^{\prime}$, and denote by $\tau^{\prime}$ the corresponding simplex in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$. By construction of $r_{\mathscr{X}^{\prime}\mathscr{X}}$, we have $\textrm{Int}(\tau_{Z^{\prime}})\subseteq r_{\mathscr{X}^{\prime}\mathscr{X}}^{-1}(\Star(\tau_{Y}))$ if and only if $h(Z^{\prime})\subseteq Y$; in this case, $h(D^{\prime}_{j})\cap Y\neq\varnothing$ for any $j=0,\ldots,q$. Thus, if $D_{\alpha}$ is an irreducible component of $\mathscr{X}_{0}$ not cutting $Y$, it follows that $h^{*}D_{\alpha}$ does not have any component along the $D^{\prime}_{j}$ for $j=0,\ldots,q$.
We deduce from this that for each $D_{i}$ component of $\mathscr{X}_{0}$ containing $Y$ and $j\in\{0,\ldots,q\}$, the coefficient of $D^{\prime}_{j}$ in $h^{*}(D_{i})$ is determined by $h$ and a local equation of $D_{i}$ in a formal neighbourhood of $Y$. This proves that $r_{\mathscr{X}^{\prime}\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on $h$ over $\widehat{\mathscr{X}_{/Y}}$.

By construction of Berkovich retraction, $\rho_{\mathscr{X}^{\prime}}$ only depends on $\widehat{\mathscr{X}^{\prime}_{/Z^{\prime}}}$ above $\Star(\tau_{Z^{\prime}})$ (see Section 1.5). If moreover $h(Z^{\prime})\subseteq Y$, then $h$ induces a morphism $\widehat{\mathscr{X}^{\prime}_{/Z^{\prime}}}\rightarrow\widehat{\mathscr{X}_{/Y}}$, hence $\rho_{\mathscr{X}^{\prime}}$ only depends on $h$ over $\widehat{\mathscr{X}_{/Y}}$.

By the independence of $\rho_{\mathscr{X}}$ on the choice of projective model $\mathscr{X}^{\prime}$ and morphism $h$, we conclude that $\rho_{\mathscr{X}}$ over $\Star(\tau_{Y})$ only depends on the formal completion $\widehat{\mathscr{X}_{/Y}}$. ∎