Local List Linear Hadwiger's Conjecture

Let $G$ be a graph, let $v(G)$ denote its number of vertices, and let $B_c[v]$ be the subgraph induced by the vertices at distance at most $c$ from $v$. For a positive integer $c$, call $G$ $c$-locally-$K_t$-minor-free if $B_c[v]$ has no $K_t$ minor for every vertex $v\in V(G)$. A graph is $k$-list-colourable if it is colourable from every list assignment with lists of size at least $k$. Local List Linear Hadwiger's Conjecture. For every positive integer $t$, there exists a constant $c_1>0$ depending on $t$, and a constant $c_2>0$, such that every graph $G$ which is $\lceil c_1\log(v(G))\rceil$-locally-$K_t$-minor-free is $c_2t$-list-colourable. This is a local list-colouring strengthening of Linear Hadwiger's Conjecture and remains open; the source gives partial results for related local colouring statements.