Linear Hadwiger's conjecture

Let $K_t$ be the complete graph on $t$ vertices, and call a graph $K_t$-minor-free if it has no $K_t$ minor. A graph is $k$-colourable if it has a proper colouring using at most $k$ colours. Linear Hadwiger's conjecture. There exists a constant $C>0$ such that for every integer $t\geqslant1$, every $K_t$-minor-free graph is $Ct$-colourable. This is a natural weakening of Hadwiger's conjecture. The best known general upper bounds are superlinear, so the conjecture remains open.