I dont have installed CSFDo you have CSF installed and with SMTP_BLOCK enabled?
R=lookuphost T=remote_smtp H=gmail-smtp-in.l.google.com [64.233.184.27] X=TLS1.2:ECDHE-ECDSA-AES128-GCM-SHA256:128 CV=yes: SMTP error from remote mail server after end of data: 550-5.7.26 This message does not have authentication information or fails to\n550-5.7.26 pass authentication checks. To best protect our users from spam, the\n550-5.7.26 message has been blocked. Please visit\n550-5.7.26 https://support.google.com/mail/answer/81126#authentication for more\n550 5.7.26 information. j3si3537758wrg.51 - gsmtpIn that case, can you search the user in the /var/log/exim/mainlog and see what is reported when attempting to send the emails?
Wordpress STILL has no native SMTP, unless you install a plugin. Why, is anyone's guess.Google is marking the mail as spam since it is being sent via phpmail rather than via SMTP.
IKR!!!!!!Wordpress STILL has no native SMTP, unless you install a plugin. Why, is anyone's guess.