For every compact $K\subset\mathcal{R}$, let $U_{s,K}$ denote the union of the regions $(s^{-1}\text{Log})^{-1}(B(x,r(x)))$ for $x\in K$; the convergence rates will be uniform on $U_{s,K}$. Notice that

$$\limsup_{s\to\infty}\frac{\text{Vol}(U_{s,K})}{\text{Vol}(X_{s})}\geq\frac{\int_{K}d\mu_{\infty}}{\int_{\partial\Delta_{\lambda}^{\vee}}d\mu_{\infty}},$$

so by taking a compact exhaustion of $\mathcal{R}$, we may assume $U_{s,K}$ occupies a percentage of the total measure arbitrarily close to 1.