It looks like the Rime api connections for the template for this example project are out of date.
When building the default project the OpenAI Realtime service works correctly but the 'custom' Rime service hits a 401.
It looks like the Rime URL and auth method for Streaming PCM are now different:
url = "https://model-7qkpx4ew.api.baseten.co/production/predict"
should be url = "https://users.rime.ai/v1/rime-tts"
"Authorization": f"Api-Key {self._api_key}",
should be Authorization: Bearer {self._api_key}