The stand has a large piece of metal inside of it to act as a weight, but this is isolated from the probe. There is no electrical contact with the probe. The metal part is probably 10-15mm from the probe body. So I'm guessing it's not doing anything.
I will try to be more specific on what test setups I've been working on. The signal generator I am using is the USB LTDZ 35-4400M spectrum analyser in CW mode. I use this because it generates a very fast 75ps edge, and is useful for evaluating the bandwidth of the probe. It outputs a fairly sensible waveform normally. I think I have also seen the same lack of effect when using the LibreVNA in CW mode, but I don't exactly remember.
Test setup 1.
No change in frequency response with or without stand/touching the probe.

Test setup 2.
Probably better ground connection between signal generator and scope.
No change in frequency response with or without stand/touching the probe.

Test setup 3.
Using different scope. USB sampling scope from SJL instruments.
No change in frequency response with or without stand/touching the probe.

Test setup 4.
Measuring frequency response on LibreVNA, with ground strap fitted. (figure 7 in datasheet).
No change in frequency response with or without stand/touching the probe.

Test setup 5.
Measuring frequency response on LibreVNA, WITHOUT ground strap fitted.
Notable change in frequency response when not holding the probe by hand.
