For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:

(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$.

(WH) If $G$ is a graph and there is no graph homomorphism $c: G\to K_\kappa$, then $K_\kappa$ is a minor of $G$.

For finite graphs, (H) is just Hadwiger's conjecture. It turns out that (H) is false for $\kappa \geq \aleph_0$: the disjoint union of all $K_\lambda$ where $\lambda < \kappa$ has chromatic number $\kappa$ but does not contain $K_\kappa$ as a minor.

However, (WH) is true for graphs $G$ with $\chi(G) \geq \aleph_0$ (see http://arxiv.org/pdf/1312.2829.pdf ). Let us call (WH) the **Weak Hadwiger conjecture**. In the finite case, (WH) translates to: "If $\chi(G) = n+1$ then $K_{n}$ is a minor of $G$." It is not known whether (WH) holds for finite graphs.

**Question**: Does (WH) imply (H) for finite graphs?