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The family has also appeared on television several times, both on local news profiles and in an episode of the ABC TV show “Wife Swap,” in which Mrs. Heene traded places with a more risk-averse Connecticut woman.

Mr. Heene denied that he rigged the balloon incident for the publicity, insisting that he had nothing to sell or advertise. The balloon was intended as part of an experiment and educational project for his sons, he said.

The Heenes, looking exhausted and frazzled, did interviews Friday morning with the three major networks. Falcon vomited during two different appearances, which Sheriff Alderden said contributed to his decision to put the interview on hold until Saturday.

If the report was a hoax, the worst the Heenes could be charged with is making a false report to authorities, a Class 3 misdemeanor. Sheriff Alderden said they could also be ordered to pay restitution for the department’s costs, a figure which has yet to be determined.