If the nano was in a metal box, as opposed to open or in a plastic one, surely some if not most of the extraneous radiation could be supressed.
I have just bought two (empty) metal cases for nano devices, one for a nano-h I got ages ago and the other for a Tinysa spectrum analyser which uses the same plastic case as the small nanovnas. I got them from one of the Chinese sites, a few weeks ago but cannot remember which one. The empty case was an option in the range of empty case to full-specced nanovna offered by this particular seller. I was actually looking specifically for cases.
The square wave is another matter however, but at least attenuators connected to Ch 0 would at least be more effective.
Steve L