Plummer–Stiebitz–Toft reformulation for independence number two

Let $G$ be an $n$-vertex graph with independence number $\alpha(G)$. Let $K_r$ denote the complete graph on $r$ vertices.

Plummer–Stiebitz–Toft conjecture. Every $n$-vertex graph $G$ with $\alpha(G)\leq2$ contains $K_{\lceil n/2\rceil}$ as a minor.

The paper states this as a reformulation of Hadwiger's conjecture for graphs with independence number at most two. A general proof is not known; the paper records a weaker $K_{\lceil n/3\rceil}$ minor result as evidence for it.