Is it possible to modify the statusCode returned by API Gateway when access is denied by a lambda authorizer?

A customer is migrating an API from Apigee to API Gateway. Currently their API returns 401 Unauthorized when a request is denied because the token is invalid. They want to maintain same status code when possible to avoid having to ask their partners to modify their code to handle different status code.

By default API Gateway returns a 403 Forbidden when the request is denied by the Lambda Authorizer. I know that is possible to provide custom messages based on $context. authorizer.key, but have not found a way to change the statusCode returned by API Gateway.