Automatic fulfillment with no product inventory

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For context, we have a Shopify application that provides merchants with an alternative shipping method and we recently released a feature where merchants have the option to have their Shopify orders be automatically marked as Fulfilled once they have created a shipment with us. 

From what I have been told, the automatic fulfillment through the API is only possible if the product inventory is available (i.e. not zero). Is this true?

If not, is there a possibility that orders can be automatically marked as Fulfilled if product inventory does not exist? For example, if the merchant does not use Shopify to track inventory for their products. 

Any help would be greatly appreciated, thank you!