Homomorphism transfer conjecture without probabilistic construction

Prove that the homomorphism-graph transfer principle yields the deterministic Ω(log(n)) lower bound for Δ-coloring acyclic graphs of maximum degree at most Δ without using the probabilistic construction of high-girth graphs in Lemma 5.12.

Background

The paper proves the lower bound by combining transfer principles with a probabilistically constructed sequence of high-girth graphs having no suitable edge-labeled Δ-coloring. It conjectures that the transfer principle itself can replace this auxiliary probabilistic lemma.