Weak Hadwiger conjecture for graphs with independence number at most two

Let $G$ be a finite simple graph, let $\alpha(G)$ denote its independence number, and let $K_n$ denote the complete graph on $n$ vertices. A graph $H$ is a minor of $G$ if it can be obtained by deleting vertices or edges and contracting edges.

Weak Hadwiger conjecture. Every graph $G$ with $\alpha(G)\leq 2$ contains

$$K_{\left\lceil\frac{|V(G)|}{2}\right\rceil}$$

as a minor.

Plummer, Stiebitz, and Toft showed that this conjecture is equivalent to Hadwiger's conjecture for graphs with independence number at most two. It remains unsolved and is presented as a possible route toward understanding, or disproving, Hadwiger's conjecture.