Kawarabayashi–Pedersen–Toft clique-minor conjecture for double-critical graphs

A finite simple graph $G$ is double-critical if it is connected and, for every edge $uv\in E(G)$, $\chi(G-\{u,v\})=\chi(G)-2$. A graph $H$ is a minor of $G$ if it can be obtained from a subgraph of $G$ by contracting edges; in particular, $K_t$ denotes the complete graph on $t$ vertices. Kawarabayashi–Pedersen–Toft conjecture. For every integer $t\ge1$, every double-critical $t$-chromatic graph contains a $K_t$ minor. This is a weaker version of Hadwiger's conjecture, which asserts the same minor conclusion for every $t$-chromatic graph. The conjecture was verified in the cited paper for $t\le7$; its general status remains open.