The Register piece mentions devices plugged in and near the mains. That suggests to me that over the air is only a concern with a very intense radiation source (mains) and only in close proximity given the inverse-square law. In the case of the fingerprint being evident because a recording device is close enough to a powerful enough EM source connected to the grid, sure, power conditioning or some other discontinuity in the wiring wouldn’t help but I think the counter measure then is simpler and more reliable.

My question was specifically about the fingerprint effect through wireline transmission and I am still uncertain that anything other than a discontinuity, like running only off of a battery (inside a Faraday cage if near the mains, of course) would be entirely effective in foiling this technique.