Does $e^{X} e^{Y} = e^{X + Y}$ imply that the operators $X$ and $Y$ commute?

Question

I am considering two self-adjoint, but unbounded operators $X$ and $Y$ on a Hilbert space. By the Baker-Campbell-Hausdorff formula, we know that $$ [X, Y] = 0 \Rightarrow e^{X} e^{Y} = e^{X + Y}.$$ But is the converse also true? If I have the property $e^{X} e^{Y} = e^{X + Y}$, can I somehow prove that the operators must commute?