Mass transport principle and measurable LCL solvability

Determine whether there exists an LCL problem solvable on all measured graphs satisfying the mass transport principle but not on general measured graphs, and likewise determine whether such a separation exists for measured graph classes locally resembling a fixed graph.

Background

Measured graphs satisfying the mass transport principle are substantially better understood than arbitrary measured graphs. The paper asks whether this structural condition provides any genuine advantage for solving locally checkable labeling problems.

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 11 (Open problems), LCL problems