Is there an infinite, simple, undirected graph $G=(V,E)$ such that there is $n\in\mathbb{N}$ with the following properties?

- $K_n$ is a minor of $G$, but $K_{n+1}$ is not a minor of $G$, and
- if $F$ is a finite subgraph of $G$, then $K_n$ is not a minor of $F$.

(Note: I use "compactness" in the title because this question has certain parallels with the compactness result of Erdös-De Bruijn.)