Λ-Cannings convergence to tree-valued Λ-resampling dynamics

Let $\Lambda\in\mathcal M_f([0,1])$ satisfy the dust-free condition

$$\int_0^1\Lambda(\mathrm{d}x)\,x^{-1}=\infty.$$

For the corresponding tree-valued $\Lambda$-Cannings dynamics $\mathcal V^{N,\Lambda}$, suppose there is a random element $\mathcal V_0^\Lambda$ such that $\mathcal V_0^{N,\Lambda}\Rightarrow\mathcal V_0^\Lambda$ as $N\to\infty$. Λ-Cannings convergence conjecture. There exists a $\mathbb U$-valued process $\mathcal V^\Lambda\in\mathcal D_{\mathbb U}(\mathbb R_+)$ with $\mathcal V^{N,\Lambda}\Rightarrow\mathcal V^\Lambda$ as $N\to\infty$; moreover, $\mathcal V^\Lambda$ is Feller and strong Markov and has the $\Lambda$-coalescent measure tree as its unique equilibrium. The conjecture identifies the infinite-population limit and its long-term equilibrium for multiple-merger genealogies.