Terraform - API Gateway with greedy path (proxy+) calling httpbin. Also includes deployment

apigateway.tf

terraform {
  required_providers {
    aws = {
      source  = "hashicorp/aws"
      version = "~> 4.0"
    }
  }
}

provider "aws" {
  region = "eu-west-1"
}

resource "aws_api_gateway_rest_api" "MyDemoAPI" {
  name        = "MyDemoAPI"
  description = "This is my API for demonstration purposes"
}

resource "aws_api_gateway_resource" "MyDemoResource" {
  rest_api_id = aws_api_gateway_rest_api.MyDemoAPI.id
  parent_id   = aws_api_gateway_rest_api.MyDemoAPI.root_resource_id
  path_part   = "{proxy+}"
}

resource "aws_api_gateway_method" "MyDemoMethod" {
  rest_api_id   = aws_api_gateway_rest_api.MyDemoAPI.id
  resource_id   = aws_api_gateway_resource.MyDemoResource.id
  http_method   = "GET"
  authorization = "NONE"
   request_parameters = {
    "method.request.path.proxy" = true
  }
}


resource "aws_api_gateway_integration" "MyDemoIntegration" {
  rest_api_id          = aws_api_gateway_rest_api.MyDemoAPI.id
  resource_id          = aws_api_gateway_resource.MyDemoResource.id
  http_method          = aws_api_gateway_method.MyDemoMethod.http_method
  type                 = "HTTP_PROXY"
  uri                  = "https://httpbin.org/anything/{proxy}"
  integration_http_method = "GET"

  cache_key_parameters = ["method.request.path.proxy"]

  timeout_milliseconds = 29000
  request_parameters = {
    "integration.request.path.proxy" = "method.request.path.proxy"
  }

}

resource "aws_api_gateway_deployment" "teststage" {
  depends_on = [
    aws_api_gateway_integration.MyDemoIntegration
  ]
  rest_api_id = aws_api_gateway_rest_api.MyDemoAPI.id
  stage_name  = "test"
}

output "api_gateway_test_url" {
  value = "${aws_api_gateway_deployment.teststage.invoke_url}/hello/world"
}

Is it possible to proxy the root, without any stages?