List 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 list assignment $L$ assigns a set of permissible colours $L(v)$ to each vertex $v$; a graph is $k$-choosable if it has a proper $L$-colouring for every list assignment with $|L(v)|\geqslant k$. List Linear Hadwiger's conjecture. There exists a constant $C>0$ such that for every integer $t\geqslant1$, every $K_t$-minor-free graph is $Ct$-choosable. The ordinary list version of Hadwiger's conjecture is false, but this linear list version remains open.