Compactness and coming down from infinity for Λ-resampling dynamics

Let $\Lambda\in\mathcal M_f([0,1])$, and let $\mathcal V^\Lambda_t$ be the tree-valued $\Lambda$-resampling dynamics at time $t$. Write $\mathbb U_c$ for the space of compact metric measure spaces. Compactness conjecture. The following are equivalent: (i) for each $t>0$, $\mathcal V^\Lambda_t\in\mathbb U_c$ almost surely; and (ii) $\Lambda$ satisfies the condition

$$\sum_{b=2}^{\infty}\gamma_b^{-1}<\infty,$$

where

$$\gamma_b:=\sum_{k=2}^b(k-1)\binom{b}{k}\int_0^1\Lambda(\mathrm{d}x)\,x^{k-2}(1-x)^{b-k}.$$

This connects compactness of the tree-valued process with the associated $\Lambda$-coalescent coming down from infinity.