I like karma, but wouldn't checking have the appearance of illegal collusion between the companies? For example, would it look like they were price-fixing?
"Oh, your honor, I was only checking whether they had an offer. We totally weren't discussing how we'd not raise our offers, with the goal of suppressing wages."
iirc there’s already an app for that in some sense, and a good amount of plausible deniability and backroom politics that can make up for any potential lawsuit. Similar issue to “proving” any kind of discrimination: a smooth corporate operator can CYA well enough to create a fake paper trail to justify any sort of decision making.
I mean, that sounds like a level of due dilegence that a lot of companies are unlikely to follow. I suspect that most don't even check the references that you give them.