AWS Lambda & API Gateway : ANY, proxy+, CORS

Terraform
1

Hello there, when creating a route_key beginning with ANY and ending with {proxy+} I can create a Lambda function to which all subroutes will point, here’s a snippet :

module "http_lambda_foo_any" {
  source             = "./lambda"
  api_id             = aws_apigatewayv2_api.http.id
  route_key          = "ANY /foo/{proxy+}"
  function_name      = "http_foo_any"
  environment        = var.environment
  lambda_bundle_path = var.lambda_bundle_path
  lambda_handler     = "src/http/foo/foo_any.handler"

For example, this matches GET /foo/, GET /foo/{id}, POST /foo/, DELETE /foo/{id}, etc.
But, CORS won’t work in this case, while it works when not using ANY and {proxy+}.
CORS configuration looks like this (values filled, of course) :

  name          = "${var.environment}-foo-http"
  protocol_type = "HTTP"
  target        = ""

  cors_configuration {
    allow_origins  = [""]
    allow_methods  = [""]
    allow_headers  = [""]
    expose_headers = [""]
  }
}

Here’s the documentation I used to do this :

Any ideas ?
Thanks.