We say that simple, undirected graphs $G, H$ are *(-1)-isomorphic* if there is a bijection $\varphi:V(G)\to V(H)$ such that for all $v\in V$ we have that the induced subgraphs $G\setminus\{v\}$ and $H\setminus\{\varphi(v)\}$ are isomorphic. (So this is a similarity notion of graphs that is weaker than isomorphism.)

For any finite or infinite graph $G = (V,E)$, we define the *Hadwiger number* by $$\eta(G) = \bigcup\big\{\alpha\in |V|\cup \{|V|\}: K_\alpha\text{ is a minor of }G\big\},$$  where $K_\alpha$ is the complete graph on $\alpha$ vertices.

**Question.** If $G, H$ are finite or infinite simple and undirected (-1)-isomorphic graphs, do we necessarily have $\eta(G) = \eta(H)$?