Biclique immersions in graphs with independence number 2

The analogue of Hadwiger’s conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{\chi (G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n\mathrm{/}2\rceil }$. We show that every $n$-vertex graph with independence number 2 contains every complete bipartite graph on $\lceil n\mathrm{/}2\rceil$ vertices as an immersion.