The Center of Mass is a defined position. It is where it is defined to be. Experiment cannot change this. An experiment that measures something else... measures something else.That does not realy answer the question and sounds cryptic.What was required was an unamabigiouse reply to the question of where exactly is the centre of mass of a spinninig gyro on a stand or string.The implication is that the CoM it is at the centre of the spinning mass but that mass is precessing about a point,"the stand".The mass moves around a point and the answer seems that the point becomes the centre of pressure rather than the CoM.
Definitions define experiments dictate.