In SOGoActiveSyncDispatcher.m line 932 there is a comment stating:
// Inside this loop we remove all the folder without write/delete permissions

I didn't look at all the code and by far I'm not able to understand it, but wht's the point of this loop?

Additional infos:
The user need all permission on the shared addressbooks ONLY at the time the EAS client is enrolled:

Steps to reproduce:
As user foo share an addressbook with user bar only granting read permissions
As user bar subscribe the folder
As user bar enroll a device with EAS
Notice that contacts in the shared addressbook are not present
Run sogo-tool manage-eas listfolders bar <androidID>
Notice that the shared folder is not present

Remove the EAS account on the phone
As user foo grant all permissions on the shared addressbook to user bar
As user bar enroll a device with EAS
Notice that contacts in the shared addressbook ARE present
Run sogo-tool manage-eas listfolders bar <androidID>
Notice that the shared folder IS present

As user foo revoke write permission on the shared addressbook
Add a contact in the shared addressbook
As user bar trigger a sync on the phone
Notice that the new contact is synced

Fixed with: https://github.com/inverse-inc/sogo/commit/f7c44863709a74b070407f6259e622bfdd8c08b2