Contact moved to private storage not visible

[WebworkrNet]

Brief background:
I overwrite identical contacts with new data and, if necessary, move this edited contact to another CardDAV account.
This was also the case here. Originally, this was a Google contact. Changing the data in “Fossify Contacts” was not the problem. The problem arose after moving the contact and then synchronizing via DAVx⁵. No matter where I moved the contact, it was not visible in the associated webmailer. From previous support cases with Bitfire, the maintainer of the sync app, I suspect that the “C flag” of the contact was not updated. To trigger it, I repeatedly made significant changes to the contact and performed further sync attempts. None of this led to success.

Finally, for testing purposes, I moved the contact to private storage. Despite the option being enabled, the private contact is not displayed. In the other cases, the contact was at least visible locally in the CardDAV account.

I suspect that the edited contact was damaged during editing or moving and therefore could not be synchronized with the respective CardDAV server at first and is now not visible among the private contacts.
All you need to do is activate the corresponding option in the Contacts app settings, right? Otherwise, I wouldn't have been able to move it there in the first place, would I?

Thank you