Lexicographic odd-cycle counterexamples

Determine whether, for an odd cycle C_{2d+1} with d ≥ 2, there exist integers m and n such that the lexicographic product C_{2d+1}[K_m] is a counterexample to H'(n).

Background

A graph is a counterexample to H'(n) when it has chromatic number greater than n while its exponential graph K_nG also has chromatic number greater than n. The paper asks whether this phenomenon occurs for lexicographic products of odd cycles and complete graphs, a particularly structured family.

References

For an odd cycle $C_{2d+1}$ for $d \ge 2$, are there integers $m, n$ such that the lexicographic product $C[K_m]$ is a counterexample to $H'(n)$?

— A survey on Hedetniemi's conjecture  (2502.16078 - Zhu, 22 Feb 2025) in Section 5, Question q5