It is illegal to knowingly and willfully make any false statement to the US government agency/officer. 18 U.S.C. § 1001, is very broad and can apply to any form you sign and submit to the US government.
And if they all lie are there the resources to track them all down, match what they say against records already out there, then prosecute and/or deport them?
Way too many of these illegal scumbags use stolen identities of citizens, living and dead, which will further muddle any process of identification.
Put the question on the Census and if it scares off a third of the illegals and they can then be identified/deported then we’re still ahead.