Return a custom header from lambda authorizer in API-gateway (HTTP api)

Question

I'm using an API-Gateway HTTP-api (not REST!) with a custom lambda authorizer. However I'd like to be able to customize the HTTP headers returned on authentication failure.
This seems possible with the REST-api (https://repost.aws/questions/QUgItZ7VFcS1iQ5wKGeU4YqA/is-it-possible-to-throw-custom-error-message-from-api-gateway-lambda-authorizer), but can't get this working with the HTTP-api.

Is this somehow possible?

Answer

Short answer is no, for now. I've tried too but the response integration is not even processed when the authorizer rejects with 403.

Answer

I am trying to add a custom lambda authorization using CFT to my HTTP api, It would be great if you can help me with it.
question: https://repost.aws/questions/QUQidcTLDQTeiEiIsMvLDvPQ/creating-a-authorizer-for-api-gatewayv2-http-api-with-a-custom-lambda-authorizer-using-cft

Return a custom header from lambda authorizer in API-gateway (HTTP api)

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