For any set $X$, let $[X]^2=\big\{\{x,y\}: x\neq y\in X\big\}$. If $G=(V,E)$ is a simple undirected graph and $e\in E$, let $G\setminus e = \big(V\setminus e, E \cap [V\setminus e]^2\big)$.

If $G=(V,E)$ is a finite graph, let $\omega(G)$ be the size of the largest clique in $G$ and let $\chi(G)$ be the chromatic number.

**Question.** Is there a finite graph $G=(V,E)$ and $e\in E$ such that $\chi(G\setminus e)= \chi(G)-2$ and $\omega(G\setminus e) = \omega(G)$?