Homomorphism to Marks’ example under non-smoothness

Prove or disprove that every Δ-regular acyclic Borel graph with no 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

Known complexity constructions use families of Borel graphs parametrized by reals. The paper proposes that excluding smooth local pieces might force a universal obstruction represented by Marks’ example, but explicitly leaves this homomorphism question open.