Weighted Aldous–Lyons soficity conjecture

Let $\alpha:\Gamma\curvearrowright (X,\mu)$ be a nonsingular action of a finitely generated group with symmetric generating system $\Sigma$. For each $\sigma\in\Sigma$, suppose

$$\frac{1}{K}\leq R(\sigma,x)\leq K.$$

Weighted Aldous–Lyons soficity conjecture. There exists a sequence of $K$-weighted graphs ${G_n,p_n}_{n=1}^\infty$ such that $\alpha$ is the limit of ${G_n,p_n}_{n=1}^\infty$.

This is the weighted version of the Aldous–Lyons “Soficity” conjecture: every suitably bounded nonsingular action should arise as the weighted Benjamini–Schramm limit of finite weighted graphs. The source presents it as the converse to the fact that every convergent sequence of weighted graphs admits a limit action; its resolution status is not specified here.