The chromatic-number conjecture for clique immersions

Let $G$ be a graph, let $\chi(G)$ be its chromatic number, and let $K_k$ denote the complete graph on $k$ vertices. Clique-immersion conjecture. If

$$\chi(G)\geq k,$$

then $G$ contains a $K_k$-immersion. This would identify the chromatic number as sufficient for a complete-graph immersion; the source presents it as open and notes that the answer to the relevant average-degree question might be $k$.