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I brought a ring (not quite that much) with me to propose to my now-fiancee in Oahu. I didn't declare it. I think 10k is the limit. I had it in my camera/computer bag (I'm sure they saw the ring in the xray).
I can't speak for most of humanity, but if I saw some guy travelling with what appeared to be a significant other and I needed to verify that it was a ring, I would bring the guy in for a personal chat away from the SO. But since we're talking TSA, all bets are off.
You can take your chances- I know 3 people personally (including myself) that have done it without issue. With the ring being only 6k, I don't think you need to declare anything unless you have more than 4k in cash as well.
Oh- bring any proof of payment with you. That's a given.
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