Mmmm..the question is..who would fund it. And if someone funded it, who would publish it...
You get funding for such a study the same way you get funding for anything else: first you run cheap small studies, perhaps in mice, perhaps observational, that show an interesting effect and motivate why you need money to run a proper clinical study.
Is this a bit? Please state your point clearly instead of conspiratorial innuendo.
There's enough money in 'crank health' that, if it were possible to publish something that might have a chance of lending some credibility (and thus expanding the breadth of the vulnerable customer base), then I would think it would have been done already.
This implies to me that the counter-evidence is strong and complete enough to discourage 'investment' into such a pursuit.