It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true?

If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

**Edit:** Addionally, what happens in $M_n(\mathbb{R})$?

**Nota Bene:** As a corollary of the counterexamples below, we deduce that if $A$ is not diagonal then $e^A$ may be diagonal.