### 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_
"x-amz-version-id",
)
if EXTRA_CORS_EXPOSE_HEADERS:
    CORS_EXPOSE_HEADERS += tuple(EXTRA_CORS_EXPOSE_HEADERS.split(","))
ALLOWED_CORS_RESPONSE_HEADERS = [
    "Access-Control-Allow-Origin",
    "Access-Control-Allow-Methods",
    "Access-Control-Allow-Headers",
    "Access-Control-Max-Age",
    "Access-Control-Allow-Credentials",
    "Access-Control-Expose-Headers",
]
ALLOWED_CORS_ORIGINS = [
    "https://app.localstack.cloud",
    "http://app.localstack.cloud",
    f"https://localhost:{config.EDGE_PORT}",
    f"http://localhost:{config.EDGE_PORT}",
    f"https://localhost.localstack.cloud:{config.EDGE_PORT}",
    f"http://localhost.localstack.cloud:{config.EDGE_PORT}",
    "https://localhost",
    "https://localhost.localstack.cloud",
    
    "file://",
]
if EXTRA_CORS_ALLOWED_ORIGINS:
    ALLOWED_CORS_ORIGINS += EXTRA_CORS_ALLOWED_ORIGINS.split(",")
ACL_REQUEST_PRIVATE_NETWORK = "Access-Control-Request-Private-Network"
ACL_ALLOW_PRIVATE_NETWORK = "Access-Control-Allow-Private-Network"
LOG = logging.getLogger(__name__)
def should_enforce_self_managed_service(context: RequestContext) -> bool: