@Peter Is CORS turned on for your storage account? If not try to add a CORS setting for your storage account. You can do this for Blob and File service tabs for all origins and methods.
Even if you have turned on CORS from the Azure OpenAI upload data screen, cross verify if it has actually add this entry in your storage account CORS settings.
After adding the above, wait for few minutes, login to your studio from a different browser or after clearing cache on your browser and retry the upload.
I hope this helps!! Thanks!!
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