Do you agree that the software engineer must be a professional programmer?

No.
Because the software engineer will get and learning everytime he or she participate. 

Yes, he/she should.
At least in one language or two, and any other knowledge in the future would be added to his/her experience.
After all, the programmer NEVER stop learning even if he/she was a prof.

Well , I prefer to put it this way : A professional Software Engineer must be a professional programmer.I think even someone who have no real experience in programming may servive as a software engineer if he has some proad knowledge in programming

OF COURSE