Derive the simultaneous DAG-exchangeable representation from the Crane–Towsner theorem

Determine whether the simultaneous representation for a collection of arrays indexed by different downsets, given by Theorem 3.2 of Jung, Lee, Staton, and Yang, can be deduced from the general representation theorem of Crane and Towsner after encoding the directed acyclic graph by equivalence relations.

Background

The paper compares the representation theorem for DAG-exchangeable arrays with the general representation theorem of Crane and Towsner, which indexes randomness by antichains of equivalence classes called “blurs.” This comparison yields a single-array representation after the DAG is encoded by equivalence relations.

Theorem 3.2 of Jung, Lee, Staton, and Yang provides a simultaneous representation for a collection of arrays indexed by different downsets. The paper states that it is unknown whether the Crane–Towsner framework can produce this stronger simultaneous representation and notes that the issue was already left open in the earlier work.