Bipartite clique minors in graphs of large Hadwiger number

The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)\ge (1+o(1))ct\sqrt{\ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $c\sim 1.276\dots$. We improve this to $h(G)\ge (1+o(1))t\sqrt{{\log }_{2}t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.