caching the result of the auth_request module

We are using nginx as a reverse proxy with the http_auth_request_module enabled. We would like to cache the requests to our authentication server in order to reduce the load of this server. This is the nginx configuration so far:

server {
  ...
  location /sso {
    auth_request /identity;
    proxy_pass http://localhost:8081;
  }

  location /identity {
    internal;
    proxy_pass              http://localhost:8081;
    proxy_cache_path        /opt/nginx/cache levels=1:2 keys=authentication:1m;
    proxy_cache             authentication;
    proxy_cache_key         $cookie_authentication;
    proxy_pass_request_body off;
    proxy_set_header        Content-Length "";
    proxy_set_header        X-Original-URI $request_uri;
  }
}

The request is being accepted and the authentication works. But nothing is being written to the cache directory. Nginx has sufficient permissions to write to the cache directory:

drwxrwxrwx   2 nobody  wheel    68B Jul 18 16:34 cache

How can I make the http_auth_request_module read from cache and make nginx write the response to cache?