Clique immersions and independence number

The analogue of Hadwiger's conjecture for the immersion order states that every graph G contains K_χ(G) as an immersion. If true, this would imply that every graph with n vertices and independence number α contains K_{⌈[Formula presented]⌉} as an immersion. The best currently known bound for this conjecture is due to Gauthier, Le and Wollan, who recently proved that every graph G contains an immersion of a clique on ⌈[Formula presented]⌉ vertices. Their result implies that every n-vertex graph with independence number α contains an immersion of a clique on ⌈[Formula presented]−1.13⌉ vertices. We improve on this result for all α≥3, by showing that every n-vertex graph with independence number α≥3 contains an immersion of a clique on ⌊[Formula presented]⌋−1 vertices, where f is a nonnegative function.