Equivalence of finitely dependent coloring constructions

Prove or disprove that the 1-dependent coloring constructed in the paper and the previously known families of 1-dependent colorings are fundamentally different, in the sense that no constant-locality randomized reduction exists between the two constructions in either direction.

Background

The paper constructs a new 1-dependent distribution over proper colorings and uses it as the basis for its quantum algorithm. It notes that 1-dependent q-colorings for q at least 4 were already known, but suspects that its construction is structurally different from prior families. The unresolved issue is whether this suspected difference persists under constant-locality randomized reductions, either from the new construction to prior ones or conversely.

References

However, we suspect that the $1$-dependent $q$-coloring presented in this work is fundamentally different from the families presented in prior work; this leads to the final open question we want to highlight here: Prove or disprove: our $1$-dependent coloring and the family of $1$-dependent colorings from prior work are fundamentally different in the sense that there is no constant-locality randomized reduction between them in either direction.

— Quantum Advantage for Distributed Symmetry Breaking  (2609.26788 - Flin et al., 22 Sep 2026) in Section "Finitely Dependent Colorings", final open question