Is there a solid example of loud overtones being generated by the "barbershop seventh" that you can point to, such as in a recording?

I don't mean to be skeptical in a negative way. It would be great if music has a built-in payoff for singing with just intonation.

I've recently heard some doubt, however, about whether this overtone effect is actually perceptible to listeners, in a way that would pass double-blind tests and stuff. It's not that the alternative is that barbershop singers are delusional or something: the alternative is that barbershop singers -- and some listeners -- have well-trained ears for just intonation, and the perception of the chord they're singing changes inside their brain when the intervals line up.

I don't doubt the power of overtones lining up. This effect is where the entire musical scale comes from, after all. But does the barbershop effect actually physically come from constructive interference (this would limit it to a small number of chords made of undertones, incidentally), or is it just a noticeable auditory illusion that occurs in trained ears?