Mass transport versus general measured graphs for LCL problems

Construct an LCL problem solvable on all measured graphs satisfying the mass transport principle but not on all general measured graphs, or determine that no such LCL problem exists; likewise resolve this question for measured graph classes locally resembling a fixed graph.

Background

The mass transport principle is central to graphings and unimodular measured graphs, but the paper notes that its algorithmic or definability advantage for LCL problems is unknown. The question is also posed for subclasses locally modeled on grids, regular trees, and related fixed graphs.

References

Is there an LCL problem that could be solved on measured graphs that satisfy the mass transport principles but not on general measured graphs? The same question is open on any subclass of measured graphs that look locally like a fixed graph, e.g., grid, regular tree etc.

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem 6.2, Section 6 (LCL problems)