Trying a test with an imaginary scenario is not going to prove much. The one thing coming through loud and clear is that there's no consensus about the origin or the exact nature of the 'bad air'. If actual problems exist, they may well be different ones in the cabin vs the flight deck.
Most suspect, in my view, is the air management hardware. It goes through a great many variations in temperature during the daily cycle, and also comes into contact with various living things be found in the air. One possibility is that relatively stable cultures of mould or bacteria are living on the inner surfaces of individual air systems, especially in temperate climates, producing germs or just noxious byproducts under certain conditions.
The DFT study seems credible enough, but it did not have specific clues or bird-dog assistance to guide the process in a particular direction. That is where flight crew taking samples on the flight deck could prove invaluable, IF they actually captured something that could be used to guide further study.
This is a case for some real diagnostic sleuthing. A couple thousand pounds for supplies and some pro bono analysis could have real effect. It just takes one corroborated "positive" sample to change the problem from "if" to "how". The ideal type of person to assist would be a University researcher with access to analytical equipment.. a chromatograph and a mass spectrometer for starters. Quid pro quo is that researcher gets material for a paper or even a thesis, and the airline industry gets a chance for some tangible evidence of an actual mechanism explaining the recurring complaints.