Inventory Managed by field

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We have an unlisted public app installed on our store, and a private app that we wrote on our store.

Now, if a product's "Inventory Managed By" field is set to our private app, the unlisted public app cannot modify inventory but our private app can. Makes sense. However, the strange thing is that if "Inventory Managed By" field is set to Shopify, the public unlisted app CAN change inventory values, but our private app cannot. How can their public unlisted app change inventory and our private app cannot when "Inventory Managed By" is set to Shopify? 

The public app has "Modify Inventory" app permissions, and our private app has Inventory: Read/Write in permissions.