Let $\mathscr{X}\subset\bar{\mathscr{X}}$ be a toric compactification of $\mathscr{X}$, i.e. a proper toric $R$-scheme containing $\mathscr{X}$ as a torus-invariant open subset. By the valuative criterion of properness, any valuation of $\mathbb{T}^{\an}$ has a center on $\bar{\mathscr{X}}$. We write $c_{\bar{\mathscr{X}}}(x)$ for the center of $x\in\mathbb{T}^{\an}$.

We start by proving that $c_{\bar{\mathscr{X}}}(\zeta(\val(x))$ is the generic point of the minimal closed torus orbit $Z$ in $\bar{\mathscr{X}}$ containing $c_{\bar{\mathscr{X}}}(x)$. We may work on the toric affine chart $\mathscr{X}_{\sigma}=\Spec R[\check{\sigma}\cap\hat{M}]$ associated with $Z$. Since the valuation $\zeta(\val(x))$ is monomial, it is enough to prove that $\zeta(\val(x))(z^{m})=v_{x}(z^{m})\geq 0$ for $m\in\check{\sigma}\cap M$ and that $\zeta(\val(x))(z)>0$ for $z$ a local equation of any torus invariant divisor containing $Z$, to have that $c_{\bar{\mathscr{X}}}(\zeta(\val(x))$ lies in $Z$. Since $z^{m}$ is regular on $\mathscr{X}_{\sigma}$, the first condition holds; the local equation $z$ is monomial and $\zeta(\val(x))(z)=v_{x}(z)>0$ since $Z$ contains $c_{\bar{\mathscr{X}}}(x)$. Moreover, we conclude that $c_{\bar{\mathscr{X}}}(x)$ must be contained in the toric interior of $Z$ by minimality of $Z$.

Now assume that $v_{x}$ is centered on $\mathscr{X}$, i.e. $c_{\bar{\mathscr{X}}}(x)\in\mathscr{X}$. Since $\mathscr{X}$ is torus-invariant and $c_{\bar{\mathscr{X}}}(x)\in Z$, we have $Z\subset\mathscr{X}$, hence its generic point $c_{\bar{\mathscr{X}}}(\zeta(\val(x)))\in\mathscr{X}$. This implies that $\zeta(\val(x))$ has center on $\mathscr{X}$. Conversely, if $\zeta(\val(x))$ has a center on $\mathscr{X}$, then $Z\subset\mathscr{X}$ and $c_{\bar{\mathscr{X}}}(x)\in Z$ as mentioned above; thus, $c_{\mathscr{X}}(x)\in\mathscr{X}$, which concludes the proof. ∎