iCloud mail not pushing inbox status

Is anyone else having the problem in iOS's stock Mail app that their iCloud mail, although set to push, won't sync the inbox status? New email gets pushed, but if I read or delete an email in my browser or on my Mac's Mail app, Mail in iOS doesn't set the email to read (or deleted as the case may be) unless I manually refresh the app or a new email comes in. I've tried Outlook and CloudMagic, and both of those sync instantly. I'd switch but for the deeper integration of Mail into the iOS ecosystem as well as the much-needed granular control over which email accounts can notify me via badge count, etc. - which only Mail provides.

If anyone has had this problem, please let me know. And if you have a solution to it, I'd be quite grateful.

Thanks in advance!