After 10-15 min, about, Plex will drop the Live TV connection. Recordings do not have this issue. Basically, the TV station selected will play and then a spinning wheel will occur where eventually it will time out. The logs will show this, although not sure if this helps or not? See image. The HDHomeRun app has no issues.
If you go over to the Live TV forum, lots of people are having issues with Live TV failing roughly 3 minutes after the start of every scheduled program in the EPG.