In the meanwhile I tested the following:
Posting to another G+ page from the same WordPress blog with the same G+ login incredentials works! The user used is “Manager” on both G+ pages.
The settings of both G+ pages used in the test above are similar. So why is it working only on one of them?
Is it right that it must be a problem on the G+ page site because posting from the same blog to antoher G+ page works?!
