(sketch) Let $x=\text{Log}_{s}(z)$ and let $m^{\prime}\in\Delta_{\mathbb{Z}}$ saturate the maximum for $L_{\lambda}(x)$. Applying $\text{Log}_{s}$ to the inequality

$$|e^{s\lambda(m^{\prime})}z^{m^{\prime}}|=|-\sum_{m\neq m^{\prime}}\frac{a_{m}}{a_{m^{\prime}}}e^{s\lambda(m)}z^{m}|\leq C\max_{m\neq m^{\prime}}\{e^{s\lambda(m)}|z^{m}|\},$$

we see

$$L_{\lambda}(x)=\langle x,m^{\prime}\rangle+\lambda(m^{\prime})\leq\max_{m\neq m^{\prime}}\{\langle x,m\rangle+\lambda(m)\}+\frac{C}{s},$$

so $\text{dist}_{\mathbb{R}^{n+1}}(x,\mathcal{A}_{\lambda}^{\infty})\leq\frac{C}{s}$. The other inequality of the claim can be proved by constructing local models of $X_{s}$ in regions whose $\text{Log}_{s}$-images are close to $x\in\mathcal{A}_{\lambda}^{\infty}$, and then use the implicit function theorem to show $X_{s}$ is a small perturbation of these local models. ∎