Lescure–Meyniel conjecture for strong immersions

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $K_{\chi(G)}$ be the complete graph on $\chi(G)$ vertices. A strong immersion of a graph $H$ in $G$ consists of an injection from $V(H)$ to $V(G)$ and pairwise edge-disjoint paths representing the edges of $H$, whose internal vertices avoid the image of the injection. Lescure–Meyniel conjecture. Every graph $G$ contains a strong immersion of $K_{\chi(G)}$. This is the strong-immersion analogue of Hadwiger's conjecture. It implies the Abu-Khzam–Langston conjecture for weak immersions, and both conjectures remain open in general; the statement is known for chromatic number at most $7$.