Well, I got most of the Zendesk inbox-bombing emails into SPAM in Gmail.
All support[at]<company>.zendesk.com were flagged, none of them reached the Inbox.
Most of whatever[at]company.tld were flagged also. I think only Headspace and another that I don't remember got to my inbox. There were some automatic SPAM flags using custom domains that are more or less known: Tinder, Squarespace, TED, ...
So I guess currently their reputation is messed up.