As of 4.0.7, a character’s account status (Alpha/Omega) is reported only for the character that is currently training. When training was switched to another character, the account status was properly reflected on that second character. The first character maintained the status as before.
Update: After the client updated the skill queues, it was found that the account status stays with the character that is training. I had switched back to training the first character and the second character lost the status.