Proof: We may assume that $X={\rm Spec}(A)$. Similary as in the proof of Proposition 4.16, there is a neighbourhood $V^{\prime}:=\{x\in X\mid s_{1}\leq|f_{1}(x)|\leq r_{1},\dots,s_{k}\leq|f_{k}(x)|\leq r_{k}\}$ of $x$ in $W$ with all $f_{a}\in A$ and real numbers $0<s_{a}<r_{a}$. We may assume that $f_{1},\dots,f_{k}$ form an affine coordinate system $y_{1},\dots,y_{k}$ on $X$. Using coordinates on $T={\mathbb{G}}_{m}^{r}$, the moment map $\varphi$ is given by analytic functions $\varphi_{1},\dots,\varphi_{r}$ on $W$ which restrict to strictly convergent Laurent series in $y_{1},\dots,y_{k}$ on $V^{\prime}$. Cutting the Laurent series in sufficiently high positive and negative degree, we get Laurent polynomials $p_{1},\dots,p_{r}$ with $|p_{a}|=|\varphi_{a}|$ on $V^{\prime}$ for $a=1,\dots,r$. By Proposition 4.16, there is a very affine open subset $U$ of $X$ such that ${U^{\rm an}}$ contains $x$ and such that $p_{1},\dots p_{r}$ define an algebraic moment map $\varphi^{\prime}:U\rightarrow T$ with ${\varphi_{\rm trop}}=\varphi^{\prime}_{\rm trop}$ on $V^{\prime}$. Choosing a neighbourhood $V$ of $x$ in ${U^{\rm an}}\cap V^{\prime}$, we get the claim. $\square$