But the brothers say they are convinced that they and their father have taken sufficient steps to create a management structure that will allow them to avoid creating the kind of appearance of conflict of interest that plagued Hillary Clinton as secretary of state while her husband continued to operate the Clinton Foundation. The measures they have taken, they say, have included explicit instructions to their domestic and international business partners not to reach out to anyone in the United States government for help.

The brothers’ expressions tightened and their voices rose when they were asked, in separate interviews, about suggestions that their father was using the presidency as a way to enhance the family’s profits.

“Who in their right mind would try to enrich themselves by spending a fortune to run against 17 seasoned politicians on the Republican side, to then go up against the Clinton machine, Wall Street, Hollywood, P.C. culture?” Don Jr. asked. “To use that as the way to enrich yourself is laughable.”

The family, he added, would face heat whatever it did. If the Trump Organization sold its assets, there would be allegations of impropriety, as foreign investors would most likely be involved. If it liquidated and put the cash into the bank, he said, his father would be accused of artificially inflating interest rates for personal gain.

For critics, though, particularly Democrats in Congress, the continuation of the global operations of the Trump Organization — even if President Trump is not directly involved — is fraught with problems, with even some Republican observers questioning whether the brothers can steer clear of trouble, regardless of their intentions.