Wheylll....
Problem is none of those solutions including Gmail/Inbox/BlueMail penetrate Doze like Aquamail does, they may use GCM in normal mode, but not in "high priority mode", i literally just tested them ALL including adding each mail app to a whitelist(ForceDoze is the app i'm using to do this) and your app is the only one that pops up, the rest only receive notifications the moment the screen comes on.
Gmail / Inbox / BlueMail are all supposedly using GCM and I'm sure they can read the docs and see what it says about "priority" mode.
And yet...
On every one of my phones with 6.0, Gmail app is often *hours* behind actual state of things -- as long as those phones remain stationary, which they do a lot.
Now, AquaMail asks to be excluded from Doze mode, maybe that's the difference.
But "priority" GCM is supposed to be "the" official solution to Doze restrictions ("here people, just rewrite your apps, pretty much from scratch, and don't forget to pay for server hosting, oh and we have this Google Cloud you could use").
https://code.google.com/p/android/issues/detail?id=197805The bug report sort of hints that this is/was a "platform" issue, not app issue.
This then is a mess -- Google puts Doze Mode in place and restricts apps, and GCM is the only way around it, the new official way, no others, and there is a bug which prevents this "official workaround" (app architecture change) from working properly?
Anyone makes mistakes, but the way they put the restrictions in place with an iron fist, before they even know if "the new way" is going to work well -- oh, I should just shut up...