Homomorphism from Marks’ example to acyclic Borel graphs

Determine whether every Δ-regular acyclic Borel graph without a smooth Borel superequivalence relation whose classes admit Borel Δ-colorings admits a Borel homomorphism from Marks’ Schreier graph of the free part of the left-shift action of Γ_Δ on 2^{Γ_Δ}.

Background

The paper observes that existing hard examples follow a common construction pattern and asks whether Marks’ example is universal under a natural exclusion of locally smooth colorable structure.