An idea: Could you post the model code? It has the form of XXnnnn.
Keep in mind that it is possible that the 70-200 f/2.8L IS II does not have one. At least some models produced a specific period did not in contrast to any other lens (See http://photo.net/canon-eos-digital-camera-forum/00Xfch)
That's the date code. Early 70-200 II's had them (mine does), then they stopped. The 70-300L does not have a date code. Seems to be the new normal?
Exactly. If users compared the date codes of their 70-200 II in combination with the existence or not of the problem there is a (remote I admit) probability that a pattern may arise.
A late lens of mine I got this year started from A (UAnnnn). Now I got 2 lenses this year (a TS-E 17 and a 35 1.4) and I do not remember which one I am talking about. I will check when I go home.