SMTP test mail works on iCloud, but not wp_mail

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I use the SMTP settings to send mails from our website with an internal plugin using wp_mail.

Sending mails from our plugin works for all mail recipients except for Apple mail addresses; like icloud.com. These mails being rejected by Apple.
But when I send a test mail from your configuration page, Apple accepts the mails.

I set a few headers for using HTML formatted mail and to set a reply-to address.