Dominating Hadwiger's Conjecture

A finite simple graph $G$ has a dominating $K_t$ minor if it contains pairwise disjoint non-empty connected subgraphs $(T_1,\ldots,T_t)$ such that, for every $1\leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. A graph is $(t-1)$-colorable if its vertices can be partitioned into $t-1$ independent sets.

Dominating Hadwiger's Conjecture. For every integer $t\geq 1$, every graph with no dominating $K_t$ minor is $(t-1)$-colorable.

Illingworth and Wood introduced dominating minors and showed that they coincide with ordinary clique minors for $t\leq 3$ but can differ substantially for $t=4$. Norin referred to this as the Dominating Hadwiger's Conjecture and suspected it was false; the paper under consideration proves that it holds for all $2K_2$-free graphs.