Proof: We choose rational functions $f_{1},\dots,f_{s}$ on $X$ with $|f_{1}(x)|=\dots=|f_{s}(x)|=1$ such that the reductions $\widetilde{f_{1}},\dots,\widetilde{f_{s}}$ form a transcendence basis of the residue field extension of ${\mathscr{H}}(x)/K$. There are rational functions $g_{1},\dots,g_{t}$ which are regular at $x$ such that $|g_{1}(x)|,\dots,|g_{t}(x)|$ form a basis of $(|{\mathscr{H}}(x)^{\times}|/|K^{\times}|)\otimes_{\mathbb{Z}}{\mathbb{Q}}$. By definition, we have $d(x)=s+t$. By (0.12) in [Du12], $f_{1}(x),\dots,f_{s}(x),g_{1}(x),\dots,g_{t}(x)$ reduce to a transcendence basis of the graded residue field extensions of ${\mathscr{H}}(x)/K$ in the sense of Temkin. There is a very affine open neighbourhood $U$ of $x$ in $X$ such that $f_{1},\dots,f_{s},g_{1},\dots,g_{t}$ are invertible on $U$. Let $\varphi_{1},\dots,\varphi_{r}\in{\mathscr{O}}(U)^{\times}$ be the coordinates of the canonical moment map $\varphi_{U}:U\rightarrow T_{U}={\mathbb{G}}_{m}^{r}$. Then the graded reductions of $\varphi_{1},\dots,\varphi_{r}$ generate a graded subfield of the graded residue field extension of ${\mathscr{H}}(x)/K$. By construction, this graded subfield has transcendence degree $d(x)$ over the graded residue field of $K$. By [Du12], Theorem 3.2, ${\rm Trop}_{U}(V)$ is a finite union of integral $\Gamma$-affine polytopes for every compact neighbourhood $V$ of $x$ in ${U^{\rm an}}$ which is strict in the sense of [Be93]. For any open neighbourhood $W$ of $x$ in ${U^{\rm an}}$, Theorem 3.3 in [Du12] shows that there is a compact strict neighbourhood $V$ of $x$ in $W$ such that ${\rm trop}_{U}(V)$ is a finite union of $d(x)$-dimensional polytopes. $\square$