People in New Hampshire have been telling visitors and immigrants from lesser states about the gold on the State House dome in Concord for years. I have myself asserted with great confidence that only those states which have had a president elected from among its citizens are eligible to have gold domes, and NH qualifies because of Franklin Pierce. A friend from Pennsylvania challenged this claim.
I can find no evidence that it is true, either by statute or custom. Counter-examples from Colorado and West Virginia reveal that there is no law which requires this, though that does not rule out custom which they choose to ignore. But I suspect this has been something which people in NH just say, finding it plausible and thinking it true, without actual basis.