The independence-number bound for clique immersions

Let $G$ be an $n$-vertex graph, and let $\alpha(G)$ denote its independence number. Independence-number clique immersion conjecture. The graph $G$ contains

$$K_{\lceil n/\alpha(G)\rceil}$$

as an immersion.

This conjecture extends Vergara's conjecture from independence number $2$ to arbitrary independence number. The bound is motivated by the inequality $\chi(G)\ge \lceil n/\alpha(G)\rceil$ and would follow from the immersion analogue of Hadwiger's conjecture; current results provide weaker lower bounds.