Ok then Steve , thats too bad it is not as simple as a loose power cord because next would come voltage monitoring to find if the main board cuts off the power on signal or the power board cuts off on his own and unless you have the correct equipment this could get complicated.
There still might be other test though but that would definitely involve removing the back cover .
Then ,did you say that both tv`s were exactly the same model ?
If so , would it be possible to get both of them at the same location for a possible board swap later on ?
Right now i`m thinking about the power board and the back cover remove and a fan cooling it down while it is playing .
And/or the opposite and heating it up with a hair dryer to see if it would provoke it into shutdown then the fan again until it can be confirm if it is a heat problem for a first thing and if so then find out which board is heat affected .
That would be the first test without meters .
My guess is that the power board is the culprit here but without being there with tools and meters ,well it would be difficult to be 100% positive.
So to resume this and to answer that question "Next ?" , well the back cover has to come off either tv