Hello everyone, hello Kostya!
I had the same problem, any widget (counter and list) got stuck loading on my Samsung Galaxy S5 with Resurrection Remix (Custom ROM - Android 7.1.2) and Nova Launcher. Push Mail is active in the app, but no mail folder uses it (might not be that clever, or is it?).
Before I added the widget (a list of the smart folder with Light "Material" Theme), I disabled the scheduled mail fetch completely - which I think could've been the problem. Indeed: I've set AquaMail to sync when rebooting and restarted my phone. The wigdet loaded properly but now it doesn't show any new mails. Even the reload button on the widget doesn't change that. To test it further, I added a five minute sync schedule and the widget got stuck loading again. The reboot helped another time, but now the reload button on the widget shows a spinning circle, which does't stop. Meanwhile I get a notification for a new mail, but it doesn't show up in the widget. Apparently, the widget has no use for me right now - but hopefully you're able to fix the problem.
Another idea: I'm using the Apps Greenify and Amplify, which control app hibernation and wakelocks. I tried to exclude AquaMail from any regulations and sync e-mail manually for power saving purposes instead. But maybe Amplify regulates some wakelocks from Aquamail anyway? I will attach a screenshot of Amplify.
Last, but not least: To really represent your awesome mail app with highly customizable settings, the widget should show mails exactly as in the app - tweaked to the user's preferences regarding background, font weight etc. of read and unread mails.