If $e^{A+B} = e^A e^B$, then $[A,B]=0$?

Question

If two matrices $A$ and $B$ commute, then $e^{A+B} = e^A e^B$ by rearrangement of the $A$'s and $B$'s in the sum. But would the converse be true?

So far I've tried to find a counterexample by considering the cases where $A$ and $B$ are 2x2 real upper triangular matrices, but those don't work.

Any hint is appreciated, thanks.