### 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_
lis for lis in default_listeners if not isinstance(lis, MessageModifyingProxyListener)
        ]
    listeners_inbound = default_listeners + listeners
    listeners_outbound = listeners + default_listeners
    data = data_bytes
    original_request = RoutingRequest(method=method, path=path, data=data, headers=headers)
    def is_full_url(url):
        return re.match(r"[a-zA-Z]+://.+", url)
    def get_proxy_backend_url(_path, original_url=None, run_listeners=False):
        if is_full_url(_path):
            _path = _path.split("://", 1)[1]
            _path = "/%s" % (_path.split("/", 1)[1] if "/" in _path else "")
        base_url = forward_base_url or original_url
        result = update_path_in_url(base_url, _path)
        if run_listeners:
            for listener in listeners_inbound:
                result = listener.get_forward_url(method, path, data, headers) or result
        return result
    target_url = path
    if not is_full_url(target_url):
        target_url = "%s%s" % (forward_base_url, target_url)
    
    if not headers.get("Host"):
        headers["host"] = urlparse(target_url).netloc
    headers["X-Forwarded-For"] = build_x_forwarded_for(headers, client_address, server_address)
    response = None
    handler_chain_request = original_request.copy()
    modified_request_to_backend = None
    
    for listener in listeners_inbound: