Thanks for replying.
I've started using Mailgun since yesterday, however not with MSMTP but relaying in postfix. (as not all domains on the server need the mails send through Mailgun)
Just in case someone is looking for a solution for this scenario and possibly also for AWS SES, I'll post how...
Same issue here, but from Cpanel to Plesk.
After resetting the mailbox password in Plesk the email clients can connect, so it looks like the passwords are not transferred, except for webmail (Roundcube).
Did you find a solution for this?