Is there a Jordan curve that has no inscribed regular $n$-gon for $n > 4$?

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I know that the Inscribed Square problem is still open. My question is, for every $n > 4$, is there a Jordan curve that has no inscribed regular $n$-gon? In fact, is there a single Jordan curve which has no inscribed regular $n$-gon for any $n > 4$?