Princess Eugenie lives in Manhattan. Will more royals be moving to America to escape public interest?

Eugenie is able to walk around NY without anyone bothering her because most Americans don't care about royals. Is it probable that Harry will some day seek the same respite in US as well? Or, is Harry too close to the throne to be allowed to live in a city where he will be afforded privacy?
Update: @Lou Brush: That's interesting. It sounds as if Ian Ball, the man who shot 3 men while trying to abduct Princess Anne, was either an American tourist or just a jolly Brit trying to say hello to the Princess and a victim of 3 men misinterpreting his intentions.
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