AWS API Gateway 429 response - no CORS headers

The 429 responses issued by API Gateway (when lowering the max concurrent requests and exceeding it) do not have the CORS headers, so the ajax request in the browser just throws error without any info that could selectively make it trigger a retry.

Any workaround but for retrying any error in the client app / i.e. any work around to have AWS include the CORS headers in the 429 response? CORS is enabled in API gateway, the OPTIONS method is there and responds the headers, and a lambda responds them itself (proxy integration) - all works well but for the 429.