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No. What "rule" might you be referring to and where did you hear about this?
Certain disclosures about green card status are already required, for example, on 1040NR and F.8840. Green card status is also a relevant consideration for various positions such as treaty tie breaker. There is little to be gained, from the IRS' perspective, by having the taxpayer disclose his/her country of origin. And unless it is codified, this may not even be legal.