Have you checked if you are currently holding the maximum 15 Synthweave Strap, Plate or Bolt? You can check how many you have stocked by entering the Appearance Customization screen. You'll see those three listed, as well as any Eververse Synthweave Templates you have.
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I checked I have no synthweaves saved up and I only have 3 synthcords and the loom still tells me inoperable