Several people here have said that bin closing should be left to the FAs. Years ago I was on a NRT-DFW flight where the FAs went around telling the pax to close the bins. I was sitting in my J seat when the FA walked by and said to me, "When you get a chance, please close the overhead bin," and then walked away. Then they made an announcement on the PA asking pax to close the bins. Was a very strange experience, and I've never had another crew like that since.
(Incidentally, I posted about this experience on FT at the time.
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