Let $G = (V,E)$ be a finite, simple, undirected graph. Hadwiger's conjecture states that

(Hadw): $K_{\chi(G)}$ is a minor of $G$.

It turns out that for finite graphs, (Hadw) is equivalent to the following statement:

(Hadw2):

If $G$ is not a complete graph, then there is a minor $M$ of $G$ such that

- $M \not \cong G$, and
- $\chi(M) = \chi(G)$.

(For an explation of the equivalence of (Hadw) and (Hadw2) in the finite case, see this.)

It is easy to see that (Hadw) fails for graphs with infinite chromatic number: $G:=\bigcup_{n\in\omega} K_n$ has chromatic number $\omega$, but $K_\omega$ is not a minor of $G$.

**Question**: Is (Hadw2) also false for graphs with infinite chromatic number?