Mail.app Never finishes "Downloading Messages"

On MacOS mail (Catalina, but also on 10.14), I’ve noticed that the client never finished downloading mail from my cPanel/WHM-based server (which has support for Apple Push for iOS, for what it is worth). The mail will come in and be readable, but it will keep saying “Downloading Messages” in the status area and the Activity window (the activity window will show it downloading, say, “Message 12 of 12.”). Because it gets stuck like this, I never receive notifications when messages come in — unless the Internet goes down and then suddenly I get a rush of notifications for everything that came in recently. I’ve noticed this problem on multiple different computers (a 2015 15″ MacBook Pro and now a 2019 16″ MacBook Pro) and on multiple networks.

With this, I am almost always seeing Mail listed as drawing significant power on the little menu bar icon for the battery.