In IMAP my be that i change something (for example delete a messages), in other mail program (Thunderbird on PC).
The situation is upgraded on K9 mail only after time configured in sincronization from profile (for me 15 minutes).
Normally in other programs if I close program and reenter, the program start with a check-mail of all profiles. Using K3b not.
Why?
Exist a button ‘check this profile now’ or ‘check all profiles now’ ?
If you have to do some tests you need to force check-mail.
Verified, is functioning (very slowly) but only when run on a profile.
In combined mode not.
In combined mode it was a good idea check all profiles.
What do you not check when you enter the program?
If you’re in the Unified Inbox and swipe down it refreshes all email accounts. Not sure if that’s what you meant by “combined mode”?
To verify, into Thunderbird on PC I have send an email to the same email address (IMAP).
Same Email configuration IMAP on K9Mail.
In Thunderbird the message arrive immediatly.
The created Email don’t arrive immediatly on K9mail.
If I’m in ‘receive combined’ (I have 3 profiles of Email in K9Mail) and swipe, dont’arrived.
If I go to the profile of the Email and do it again arrive.
Still not sure what “combined mode” means. Could you explain more in detail or send a screenshot to understand this better?
I have changed language to English (was Italian).
‘Unified inbox’
You seem to have enabled Poll with 15 minutes period. You should set to Pull and disable Poll.
On mobile phones closing an app is not necessarily closing it completely.
BTW: I guess K3b is a typo, should be K9?
Oppps … sorry The Linux spirit in me emerges 
How?
I would like sync all but is not simply undestood how:
Fetching mail I have:
Sync messages any time (no limit)
Fetch messagess up to 32 Kb
Folder Pool 15 minutes
Poll folders All
Push folders Only 1st class
Max folders to check with push 50
Refresh IDLE connection 24 minute.
And now?
Here I see that you have set polling to 15 minutes (as expected as you wrote that it will synch after 15 minutes).
Poll folders you have set to All - this means it will fetch all folders you have after that synch time.
Push folders you have set to use 1st class only. Means that it will get push only for those folders you have configured to be 1st class.
As you have that difference between poll folders and push folders I assume that you did not configure all folders you are interested in to get push notifications as 1st class.
Check on “Folder Settings” the configured classes for each folder.