Factor-of-iid Δ-edge coloring of the regular tree

Construct, or prove the impossibility of, a factor-of-iid edge coloring with Δ colors on the Δ-regular tree T_Δ, equivalently determine whether measurable König line coloring holds for the iid graph on T_Δ.

Background

A factor-of-iid perfect matching is known on the Δ-regular tree, whereas measurable König line coloring fails for general measurably bipartite acyclic measured graphs. The paper asks whether the highly symmetric iid regular-tree setting nevertheless admits the optimal Δ-edge coloring.

References

Is there a factor of iid edge coloring with $\Delta$ colors on the $\Delta$-regular tree $T_\Delta$? In other words, does measurable K\H{o}nig's line coloring theorem hold for the iid graph on $T_\Delta$?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem immediately following the discussion of measurable perfect matchings, Section 6 (Edge colorings)