The app syncs a certain number of recent messages per folder, starred or not.
What you can do, and it should help
somewhat is to increase how many that is (the defaults are rather conservative).
Long press the account -> options and folders -> account options -> increase "messages to sync".
The larger the setting, the more work the app has to do, keeping that many messages "in sync" with the server (read/unread/starred state, checking if having been deleted) -- so please don't jump into "thousands" range at once