Good question. I would also forward this post link to Efficy cloud mailbox.
Efficy supports for the Windows Authentication the following providers: NTLM, Kerberos. Recently authentication using SAML 2.0 was added as well. I never used it in a project however, so I don't have usage experience.
I believe the Kerberos and NTLM are complicated and refused for cloud customers, but maybe SAML 2.0 can be used?
Interested to see the feedback from cloud...