Odd minors or odd immersions in graphs with independence number two

Kühn, Sauermann, Steiner and Wigderson recently disproved the Odd Hadwiger Conjecture, even for graphs with independence number 2. For this class of graphs the conjecture is known to be equivalent to the following: every $n$-vertex graph $G$ with independence number 2 contains $K_{\lceil \frac{n}{2}\rceil }$ as an odd minor. While this does not hold, we prove that every graph $G$ with independence number 2 contains $K_{\lceil \frac{n}{2}\rceil }$ as an odd minor or as a totally odd immersion.