What email provider for Monitored Inbox

What email service yall use for the monitored inbox? I am on hostgator and the bounces and complaints are piling up in the inbox, but mautic does not seem to be monitoring it. The peeps at hostgator say that the see where the login is successful – but that mautic’s attempt to get past the certificate on a shared host is somehow creating an issue because they use a wildcard certificate. In mautic when I test the credentials I get a “Success!” but it’s not actually monitoring the inbox or updating my contacts. My SMTP is going well with AMazon AWS. Any suggestions? Should I set up a hotmail bounce inbox or something of the sort? Thanks!
Alaina