Frequency of large-girth counterexamples to H'(n)

Determine the probability that a random graph G of sufficiently large girth and chromatic number greater than n is a counterexample to H'(n), namely that χ(K_n^G) > n.

Background

The paper discusses the absence of a clear structural explanation for known counterexamples to H'(n), where H'(n) asserts that χ(G) > n implies χ(K_n^G) ≤ n. It asks whether counterexamples are abundant among random graphs of large girth and chromatic number exceeding n.