For any finite, simple, undirected graph $G=(V,E)$ denote by $G_2 = (V_2, E_2)$ the graph, in which $V_2$ and $E_2$ are defined as follows:

$V_2 = \big(V\times\{1\}\big) \cup \big(V\times \{2\}\big)$, and

$E_2 = \big\{\{(x,i),(y,i)\} : \{x,y\} \in E \textrm{ and } i \in \{1,2\}\big \} \cup \big\{\{(v,1), (w,2)\} : v,w\in V\big\}$.

In informal terms, $G_2$ consists of two copies of $G$ such that every vertex in one copy is connected to every vertex in the other copy.

Is there an example of a graph $G=(V,E)$ such that there is $n\in\mathbb{N}$ such that

- $K_n$ is no minor of $G$, but
- $K_{2n}$ is a minor of $G_2$

?