Chromatic number implies clique immersion (Lescure–Meyniel conjecture)

Prove that for every integer k ≥ 1 and every graph G, if the chromatic number χ(G) ≥ k, then G contains an immersion of the complete graph K_k.

Background

The parameter imm(k) is the minimum average degree guaranteeing a K_k-immersion; current bounds show imm(k) = O(k). Strengthening this to depend on chromatic number would unify and improve degree-based results. The conjecture is widely open for general k and would follow if minimum degree k guaranteed a K_k-immersion.