Jiménez–Quiroz–Thraves Caro totally odd immersion conjecture

For a graph $G$, let $\operatorname{toi}(G)$ be the maximum integer $t$ such that $G$ contains a totally odd strong immersion of $K_t$, and let $\chi(G)$ denote its chromatic number. Jiménez–Quiroz–Thraves Caro conjecture. For every graph $G$, we have

$$\chi(G)\leq \operatorname{toi}(G).$$

This extends the immersion–chromatic number conjecture to dense graph classes, including complete bipartite graphs, for which the ordinary immersion parameter can behave differently. The source presents it as an open conjecture and cites inspiration from Churchley’s work.