And I definitely wouldn't put much weight on my failure to find a proof—I'm still a student, so my research skills still leave much to be desired :-)

@PavelShuhray Personally, I’m not aware of anyone who’s written up a proof of this specific statement. And unfortunately, I haven't yet been able to prove it myself. However, it has been a while since I last thought about the problem, so I should probably take another shot at it!