Borel chromatic-index complexity on acyclic regular graphs

Determine whether, for every Δ≤k≤2Δ−2, distinguishing Borel chromatic index k from k+1 on Δ-regular acyclic Borel graphs is Σ¹₂-complete.

Background

Existing work establishes Σ¹₂-completeness for distinguishing chromatic index Δ from Δ+1 in bounded-degree Borel graphs. Since acyclic Δ-regular Borel graphs can realize intermediate chromatic indices, the paper asks whether the same maximal complexity holds at every intermediate threshold.

References

Is it $\mathbf{\Sigma}{1}_2$-complete to distinguish between any $k$ and $k+1$ Borel chromatic index on $\Delta$-regular acyclic graphs, where $\Delta\le k\le 2\Delta-2$?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem immediately following the discussion of Borel chromatic-index complexity, Section 6 (Complexity)