The direct implication follows from the continuity of $\operatorname{Log}_{\mathcal{V}}$. For the reverse implication, assume that $\lim_{t\to 0}(\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}=\mu_{0}$ and consider the following three subsets of $C^{0}({\mathcal{V}})$: $A_{1}$ is the set of functions of the form $\operatorname{Log}_{\mathcal{V}}^{*}\varphi$, where $\varphi\in C^{0}(\Delta({\mathcal{X}}))$; $A_{2}$ is the set of functions of the form $\pi^{*}g$, where $g\in C^{0}({\mathbb{D}}_{r})$; and $A_{3}=C^{0}_{c}({\mathcal{V}}\setminus\Delta({\mathcal{X}}))$ together with the constant function 1. Then the real vector space $A\subset C^{0}({\mathcal{V}})$ spanned by functions of the form $f_{1}f_{2}f_{3}$, with $f_{i}\in A_{i}$ is easily seen to be an ${\mathbb{R}}$-algebra that separates points and contains all constant functions. By the Stone-Weierstrass Theorem, $A$ is dense in $C^{0}({\mathcal{V}})$, so it suffices to prove that $\lim\int f\mu_{t}=\int f\mu_{0}$ for $f\in A$. By linearity, we may assume $f=f_{1}f_{2}f_{3}$ with $f_{i}\in A_{i}$. We may further assume $f_{3}=1$. Write $f_{1}=\operatorname{Log}_{\mathcal{V}}^{*}\varphi$ and $f_{2}=\pi^{*}g$. Then

$$\lim_{t\to 0}\int_{X_{t}}f\mu_{t}=\lim_{t\to 0}g(t)\int_{X_{t}}\varphi\circ\operatorname{Log}_{\mathcal{V}}\mu_{t}\\ =\lim_{t\to 0}g(t)\int_{\Delta({\mathcal{X}})}\varphi\ (\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}=g(0)\int_{\Delta({\mathcal{X}})}\varphi\mu_{0}=\int f\mu_{0},$$

which completes the proof. ∎