But https is not normal.
It's encrypted traffic and can indeed fail. I fear I've failed you here. You need to talk directly to the carriers and hear from them if this should or should not work. I'm just a designer, programmer and lead on some designs where we used data over cellular and came away a little wiser and a little more gray.
Let me be a little more blunt. You would provide more than one way to get email. For example, Outlook via the usual pop3/smtp as well as some https web page.