Updating your listing inventories requires you to have a Shop Uploader template filled out. Use the Templates page to create a new template or to download a baseline.
Once created, make sure the listing_id column is populated. When left empty, none of your listings will be updated.
After filling out the listing_id column, set the action column to inventory. The inventory action only looks at the quantity column and updates your listing inventories. Additionally, this makes it so that you use Usage (inventory) listings instead of Usage (create/update) listings.
With this all set up, you can now upload your template to update your listing inventories.