Can't provide “proof” other than 60+ years of living in wetback land, and to say I don't believe any of the studies mentioned are close to correct. The problem is, the studies are always based on the obvious and visible targets. They never get to the total effect.
I'm not about to get sued, but you would be surprised how many large companies here, and probably everywhere else benefit from illegals. They have all sorts of ways to camouflage it, but it still happens. A few of them even go so far as to brag how they only deal with companies that check the "record of all their employees." That just means that the company they are contracting with in turn subcontracts to another company that has all the wets. How much of that shows up in the price to the consumer and how much of it just helps the bottom line of the company I wouldn't know. But it is everywhere.