Mass transport advantage for LCL problems

Determine whether there exists an LCL problem solvable on all measured graphs satisfying the mass transport principle but not on general measured graphs, or analogously on a subclass of measured graphs locally resembling a fixed graph.

Background

The paper observes that graphings and other mass-transport-preserving measured graphs are more extensively studied than arbitrary measured graphs, but no separation for LCL solvability is known.

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, Section 7 (LCL problems)