FaceTime with Continuity Camera keeps stopping

I've used both of our iPhone 14 Pros in Continuity Camera mode, using the FaceTime app on a current model Apple TV with the same results each time. The FaceTime call drops after several minutes each time, forcing us to reconnect - only to have the call drop a few minutes later.

The Apple TV is networked via ethernet. The video quality and Center Stage have been awesome but the video calls appear to time-out after several minutes (5-10 minutes) every time. Is anyone else experiencing this phenomenon?

[Re-Titled by Moderator]