$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Question

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $B$ are bounded, then I know that this must be the case. However, I am not sure if the same must be true for unbounded operators. Does anyone have a proof or a counterexample?