Atomlessness of tree-valued Λ-resampling dynamics at almost every time

Let $\mathcal V^\Lambda=(\mathcal V_t^\Lambda)_{t\geq 0}$ be the tree-valued $\Lambda$-resampling dynamics. Atomlessness conjecture. For Lebesgue almost every $t$, the metric measure space $\mathcal V_t^\Lambda$ has no atoms. Although a reproduction event can create an atom in the population measure, subsequent tree growth is conjectured to destroy that atom immediately, yielding atomlessness at almost every time.