In over 6 years of using Mailbox, I only remember one website that rejected the domain. It's a government website and they only started rejecting "invalid" (non-duopoly) domains sometime this year.

I do occasionally get emails that take longer to arrive into my inbox (between 5 minutes up to 1 hour), but the emails always arrive eventually. Or maybe I haven't noticed...

Do you remember which websites rejected your Mailbox address?

I've been on mailbox for 6 years and I think the only issue I've had with rejections has been the email confirmation from some Discourse-based forums. But after I contacted the hosts and was added manually, the forums' emails made it through with no issue