Churchley's totally odd immersion conjecture

Let $G$ be a graph. Write $\chi(G)$ for its chromatic number. A totally odd immersion is an immersion in which every path representing an edge of the immersed graph has odd length. Let $K_t$ denote the complete graph on $t$ vertices.

Churchley's conjecture. Every graph $G$ with $\chi(G)\ge t$ contains a totally odd immersion of $K_t$.

This strengthens the immersion analogue of Hadwiger's conjecture and is incomparable with Hajós's subdivision conjecture. The paper proves it for line graphs of constant-multiplicity multigraphs, while the general assertion remains open.