The soficity conjecture for minor-excluded unimodular random rooted graphs

Let $H$ be a finite graph, and let $(G,\rho)$ be a unimodular random rooted graph. Assume that, almost surely, $G$ does not have $H$ as a minor.

Minor-excluded soficity conjecture. Every such unimodular random rooted graph $(G,\rho)$ is sofic.

The paper's proof strategy establishes the result under a one-endedness assumption, using filament decompositions and approximation by graphs of bounded treewidth. The conjecture asks whether that assumption can be removed; the source says that the authors believe this should be possible.