There are reporting agencies that can be used to determine who employs a person. That's not including some other shady shit I wouldn't put past the company getting up to (i.e., bribing recruiting firms to tattle on former employees). Plus, I bet there's clawback provision if you don't tell them and they find out about it.

But yeah, I'd take that gamble and just keep a low profile online. Or one could slow walk the hiring process: land a job then push your start date back out 61-x days.