You installed the DeepL plugin from the integrated AppStore and the plugin and Studio versions seem compatible.

Your key is valid and should be working without any problems. However, after clicking "OK" in the DeepL plugin to verify the license key, a new, empty line appears right below (see the "before" and "after" screenshots) as if there was an error message, but with no text:

Before:


After:

Change the proxy authentication settings in the Studio config file as advised here. Please note that you need admin rights for this.

When it's done, you should be able to connect the DeepL plugin to the DeepL server.