Trying not to get my hopes up but.. .

Most proof dies show an MDO, this is the second impression that you are seeing.
Proof die have 2nd die pressings.

 

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It's the proof coins that are struck twice (at least). https://www.usmint.gov/learn/production-process/coin-production
Proof dies get special polishing, frosting, etc, but not a second pressing (at least not in the single-press era; previously all dies were hubbed at least twice). https://www.usmint.gov/learn/production-process/die-making
Correct me with a source if I'm wrong, but that's the way I've always understood it.

 

Thank you all for the information. I appreciate everyone taking the time to respond.