Tree-chromatic weakening of Hadwiger's conjecture

Let $G$ be a graph and let $t$ be an integer. The tree-chromatic number $\mathsf{tree}\textnormal{-}\chi(G)$ is the minimum, over tree-decompositions of $G$, of the maximum chromatic number of a bag. Tree-chromatic Hadwiger conjecture. If $G$ is a graph without a $K_{t+1}$-minor, then $\mathsf{tree}\textnormal{-}\chi(G) \leqslant t$. This is proposed as a weakening of Hadwiger's conjecture and is open; the paper proves it for $t=5$ without using the Four Colour Theorem.