### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

Authorization headers being sent to API gateway are received on lambdas as "empty string"

### Expected Behavior

_No response_

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

Run a API Gateway with lambda integration

### Environment

```markdown
- OS:
- LocalStack:
```


### Anything else?

_No response_
self, method: str, path: str, data: MessagePayload, headers: Headers
    ) -> Optional[RoutingRequest]:
        return RoutingRequest(
            method=method,
            path=self._adjust_partition_in_path(path, self.DEFAULT_INBOUND_PARTITION),
            data=self._adjust_partition(data, self.DEFAULT_INBOUND_PARTITION),
            headers=self._adjust_partition(headers, self.DEFAULT_INBOUND_PARTITION),
        )
    def return_response(
        self,
        method: str,
        path: str,
        data: MessagePayload,
        headers: Headers,
        response: Response,
    ) -> Optional[RoutingResponse]:
        
        if is_internal_call_context(headers):
            return None
        response = RoutingResponse(
            status_code=response.status_code,
            content=self._adjust_partition(response.content),
            headers=self._adjust_partition(response.headers),
        )
        self._post_process_response_headers(response)
        return response
    def _adjust_partition_in_path(self, path: str, static_partition: str = None):