It might make some difference but when you read about Social Security employees selling SS numbers and the holders' info why wouldn't the supposed uncounterfeitable cards turn up in the hands of illegals.
Sure some numbers are stolen, but I suspect a lot of folks are just making up numbers that fit the known sequences. Employer gets a letter from Social Security, the employee bails.
By 1989 I had four different illegals using my SSI number. I don't know what the number is today, probably 20.