The strong odd Hadwiger-type conjecture for clique immersions

Let $t\in\mathbb{Z}^+$ and let $G$ be a graph. An immersion of $K_t$ is strong odd if its paths are pairwise edge-disjoint, have odd length, and no terminal is an interior vertex of a path. Strong odd immersion coloring conjecture. If $G$ has no strong odd $K_t$-immersion, then

$$\chi(G)<t.$$

This conjecture extends clique-immersion coloring conjectures by imposing both the strong and odd conditions. It is known for $t\leq 4$ and for several special graph classes, while the general statement remains open; the paper's main theorem gives a bound when $\alpha(G)\leq 2$.