When auxiliary.<task> config has base_url set but api_key is empty
(common when user expects env var fallback), _resolve_task_provider_model()
returned provider="custom" with api_key=None. This caused downstream
client construction to make API calls without an Authorization header,
resulting in HTTP 401 errors.
Fix: only return "custom" when BOTH cfg_base_url AND cfg_api_key are
non-empty. When base_url is set without api_key but with a known
provider (e.g. "openrouter"), pass through to that provider so it can
resolve credentials from environment variables.
Fixes#16829
vominh1919
96514de472