Never heard of that. Your clock draws power continuously from the battery, why doesn't that cause the PCM to stay "on".
Most modern cars have 'keep alive memory' that needs to have some current draw such as learned settings in the PCM etc.
You can check your key-off battery drain using a meter. Some modern vehicles will 'step down' their key off drain after certain increments of time.
I'm really skeptical that having an LED glowing would cause the PCM to stay 'on'. This would suggest that no one could install one of those aftermarket alarm systems with the flashing LED light.
Plus, this flies in the face of almost everyone elses experience with leaving ipods, cell phones and the like in the car 24/7.
In fact, I left a Wagan inverter in my car for three years, turned on with the cooling fan running 24/7 (which I guarantee drew more power than an LED) and it wasn't until I accidentally left my 12v cooler on that my battery finally died.
Somewhere there are probably some key-off power drain specifications for the KJ.