That may be your case with actual key. Those with push button start, have found this to be the case a large percent of the time. He has a Lariat; $15 dollars is cheap enough to try.
Get a Faraday box for your keys, I had the same issue. Truck not going to sleep "so to speak". The key communicates with the truck. Solved my issue and no longer get notifications that acc. have been shut down to conserve battery.