Yes, I've seen your message:
https://groups.google.com/d/msg/k-9-mail/A_cF14avnzg/xsr_n1LpBwAJThat's what Doze Mode is, by design.
It turns off scheduled events, "wake locks", and network connectivity, this is how Google made it work, intentionally.
They don't want apps "running" in Doze mode -- they want apps to work off of Google Cloud Messaging.
And they have no trouble enforcing this at system level, by "enforcing" I mean that an app can try (to call the functions), but it won't have any effect.
To a mail app, it means it has to be "cloud based" like TypeMail, MyMail, Outlook, passing your messages through their servers (not every user's cup of tea, but surprisingly few are even aware).
Gmail app will obviously work fine, with @gmail accounts, but I believe that IMAP and Exchange accounts will be affected by Doze mode too.
There are some ways to get around it, but it's very limited (e.g. scheduled events won't be fired more often than once per 15 minutes).
Personally, I'm watching the K9 discussion with great interest, to see what sort of "story" they come up with for their users.