A shipping date of June 27 would have been readily achievable following the documented overdub session on June 16, given that only the third side had to be remastered, and the first disc of each copy could already have been pressed.  The sleeves could have been printed up for months.  A shipping date of June 20, on the other hand, would have been very difficult to meet. 

A June 27 shipping date would also be consistent with the June 29 Columbia press release shown below, announcing the album's release; after all the delays, the company probably didn't want to put this out until they knew the album was actually in distribution.  This press release was evidently used as the basis of Cash Box's July 9 news item as seen in an earlier note.

Columbia press release Jun 29 1966

Reproduced from Issue 50 of UK fanzine The Telegraph, 1994.  If anyone reading this has access to the second page of this press release, please do get in touch!