I just got a 2016 LWB wagon with a factory installed tow package (yes it is listed on the window sticker) and it does this exact same thing. At first I thought there was a fault in my trailer, but after trying different tests found that it does not matter what is plugged in. I plan to look it over when I have some time and see if there are any obvious wiring problems before trying the warranty option.
It is puzzling that this thread was started in 2010 and the problem has surfaced on a 2016 unit. Maybe not a design fault or defect, but just a mistake in factory installation ? Like a bad ground or something ?