I have a total of 5 accounts in K9, one on one server and four on another server. Push only works on the single server account and on the first account of the second server. (The second server can have several different DNS names or all accounts use the same DNS name but they all point to the same IP Address. Changing the name of the server for the different accounts makes no difference.)
This is not random — it’s always the same two accounts where Push works and the rest of the accounts only fetch mail when I do a pull down refresh.
It’s been like this for years now. I keep hoping an update will fix it but it hasn’t happened. I realized I’d better report it if I want it to be fixed.