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 $K_{χ(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n/2 \rceil}$. We show that every