If $G=(V,E)$ is a finite, simple, undirected graph, then by $\eta(G)$ we denote the maximum integer $n\in \mathbb{N}$ such that $K_n$ is a minor of $G$. If $e\in E$ we write $G\setminus e$ to denote the graph $(V, E \setminus \{e\})$.

Is there a finite graph $G=(V,E)$ and $e\in E$ such that $\eta(G\setminus e) < \eta(G)-1$?