The totally odd strong immersion conjecture

Let $G$ be a graph. Write $\chi(G)$ for its chromatic number. A totally odd strong immersion is a totally odd immersion whose terminals do not occur as interior vertices on the paths representing edges. Let $K_t$ denote the complete graph on $t$ vertices.

Totally odd strong immersion conjecture. Every graph $G$ with $\chi(G)\ge t$ contains a totally odd strong immersion of $K_t$.

This is presented as a strengthening of Churchley's conjecture, motivated by the paper's proof for line graphs of constant-multiplicity multigraphs. The general conjecture remains open.