Hi Gerhard,
Sorry about missing your response, I didn't get notification on your previous response somehow. Unfortunately, we don't have a reproable set of steps yet.
For the one Exchange account that mail doesn't sync automatically, do you know what Exchange server version you are connecting to? When you say 'no activity', are you saying you don't see any status communication or are you not getting mails and know it's not working?
If it's just not syncing at all - can you send me a private message and I'll walk you through turning on error logging so you can send logs to us to troubleshoot?
Thanks,
Sunder