I tested this when my dad got one of the push-button start cars. Doesn't work, at least not in his Acura.

Give it a few minutes and it will start to complain, and then you will lose engine, but not electrical power.

Doesn't help on any keyless car I've ever used - the car will beep at you when the key leaves the vehicle, but the engine won't shut off until you hit the stop/start button. And in this case, that button apparently did not work.