Pick $\chi\in C^{0}_{c}({\mathcal{U}})$. If $\kappa_{0}=0$ and $q=d$, then

$$\lim_{t\to 0}(\operatorname{Log}_{{\mathcal{U}}})_{*}(\chi\mu_{t})=\left(\int_{Y^{\prime}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)\right)b_{\sigma^{\prime}}^{-1}\lambda_{\sigma}^{\prime}$$

in the weak topology of measures on $\sigma$, with $\sigma^{\prime}$ the unique $d$-dimensional face of $\Delta({\mathcal{L}})$ contained in $\sigma$. Otherwise (i.e. if $\kappa_{0}>0$ or $q<d$) $(\operatorname{Log}_{{\mathcal{U}}})_{*}(\chi\mu_{t})\to 0$.