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What does "Don't put your nose down" mean?
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I've never heard that expression in 80 years as a native English speaker.

Where did you encounter it?

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cobra scarf 490 What does "Don't put your nose down" mean?

I've never heard of it either.

I have heard this one:

Don't look down your nose at someone.

https://www.collinsdictionary.com/us/dictionary/english/to-look-down-your-nose-at-someone