Arrogant as a Mount Rushmore president? (9)

I believe the answer is:

bigheaded

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'arrogant' is the definition.
Both the answer and definition are adjectives. Perhaps you can see an association between them that I can't see?

I can't explain the rest of the clue.

Can you help me to learn more?

(Another definition for bigheaded that I've seen is " Conceited with large cranium".)

I've seen this clue in The Sunday Times and The Times.

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