If the user/organization fields used to be updated in Zendesk according to the app's settings, but this is no longer the case, you have to make sure that some fields have not been recently deactivated.
Indeed, if you were using a ticket field in one of your templates that has been deactivated since, the app will no longer be able to make the link between this deactivated ticket field and its use in the template. This will cause the ticket update to be blocked.
In the example above, I am using the ticket field "Street address" in my template but this ticket field has been disabled in my instance.
As a result, the app will not be able to update my tickets until the disabled field is removed from my template.