  In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is
$(t-1)$-colorable for every $t\ge 1$. While Hadwiger's conjecture does not hold
for list-coloring, the linear weakening is conjectured to be true. In the
1980s, Kostochka and Thomason independently proved that every graph with no
$K_t$ minor has average degree $O(t\sqrt{\log t})$ and thus is $O(t\sqrt{\log
t})$-list-colorable. Recently, the authors and Song proved that every graph
with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta >
\frac 1 4$. Here, we build on that result to show that every graph with no
$K_t$ minor is $O(t(\log t)^{\beta})$-list-colorable for every $\beta > \frac 1
4$.
  Our main new tool is an upper bound on the number of vertices in highly
connected $K_t$-minor-free graphs: We prove that for every $\beta > \frac 1 4$,
every $\Omega(t(\log t)^{\beta})$-connected graph with no $K_t$ minor has $O(t
(\log t)^{7/4})$ vertices.
