Linear-size totally odd immersion conjecture

Let $G$ be a finite, undirected, loopless graph, and let $\chi(G)$ denote its chromatic number. A graph $H$ is a totally odd immersion of $G$ when the edges of $H$ are represented by pairwise edge-disjoint odd-length paths in $G$, with distinct terminals that are not internal vertices of those paths. Linear-size totally odd immersion conjecture. There exist $\epsilon>0$ such that every graph $G$ contains $K_{\lceil\epsilon\chi(G)\rceil}$ as a totally odd immersion. This is presented as a weakening of the preceding conjecture and is motivated by the totally odd immersion theorem for graphs whose zigzag number equals their chromatic number; its general validity remains open.