Hi Grondeaux,
Thanks for contacting Apple Support Communities!
We understand that you're unable to maintain your AirPlay connection from iTunes.
You may consider verifying that you don't have any wireless interference that may impact your connection as outlined here: Resolve Wi-Fi and Bluetooth issues caused by wireless interference - Apple Support
Also, check out the additional troubleshooting steps outlined in this article for help with AirPlay: If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support
"Still need help?
If you have another issue using AirPlay, learn what to do.
If your music unexpectedly stops
If your music stops playing unexpectedly, use your device or Siri to start playing music again. In some situations, the Home app might be using your HomePod,* Apple TV, or AirPlay-compatible speaker as the main audio source for streaming music (even if your HomePod* or Apple TV isn’t currently playing music). Using Siri on that device or performing another task might cause playback to stop everywhere. You can also use an iOS device or computer with iTunes as an audio source to manually stream music instead.
If you see video but don't hear audio
If you don't hear audio when streaming video from your iOS device or Mac to your Apple TV, make sure that the volume is turned up and not muted on both your iOS device and your television or receiver.
If you're streaming video from an iPhone or iPad, check the Ring/Silent switch. If it's set to silent, you'll see an orange line. Toggle the switch to turn on the ringer.
If you still can’t use AirPlay after trying these steps, contact Apple Support."
We hope this helps you.
Take care!