There is no supporting evidence. That is part of the problem. The FAA has enough on their hands trying to get the air traffic control system into the latter half of the twentieth century, without trying to test for every stray signal and how it might affect the navigation. A blanket policy, restricting all devices does that.

The wiring is shielded to an extent but there is no special safe room for the avionics. A lot of it is in the nose, right in front of the crew. It is, after all, a very thin tube of aluminum we are all shooting around in. Weight is something to be avoided.

So how do the signals get to the avionics? Let's say that there is an antenna outside the avionics room. It needs to pick up signals. All of these devices emit signals, and some of the signals will go outside the airplane to said antenna.