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After the Factions mode update, I've been losing in-game audio. This has lead me restart the application in order to get audio back. PS Party audio is maintained and consistent throughout.
Please check your Windows sound settings to make sure you have everything hooked up correctly. Also, in Options->Sound check to make sure the audio output device is the correct one for your system.
If trying the above does not work then try this:
On the Warframe launcher, click the little gear icon on the top right. In the Download Cache section hit "Verify" and "Optimize".