Totally odd immersions in line graphs

The immersion-analogue of Hadwiger’s Conjecture states that every graph $G$ contains an immersion of $K_{\chi (G)}$. This conjecture has been recently strengthened in the following way: every graph $G$ contains a totally odd immersion of $K_{\chi (G)}$. We prove this stronger conjecture for line graphs of constant-multiplicity multigraphs, thus extending a result of Guyer and McDonald.