Woodall's chromatic conjecture for complete bipartite minor-free graphs

Let $G$ be a graph, let $\chi(G)$ be its chromatic number, and write $G \not\succeq K_{s,t}$ when $G$ has no $K_{s,t}$ minor.

Woodall's chromatic conjecture. For every $s,t \in \mathbb{N}$, if $G$ is a graph with $G \not\succeq K_{s,t}$, then

$$\chi(G) \le s+t-1.$$

This conjecture is a weakening of Hadwiger's conjecture and was independently proposed by Seymour. The source describes it as open, while noting that Woodall proved the corresponding list-colouring conjecture when $s \le 2$.