What would be the atmospheric pressure gradient in an open pit reaching to the centre of the moon?

Inspired by this question, what would the pressure stabilize at over tens of thousands of years (at the moon's centre, surface, and any other interesting points) if you created a structurally stable pit (built of unobtanium) reaching to the centre of the Moon, and filled it with the same gases found at sea level on earth?

Assuming such atmosphere would be actively stabilized to ~20Â°C where the pressure was at least 13 kilopascals, and the top end of the pit is subject to the same boiling-off any previous Moon atmosphere was.

The perceived gravity at the bottom of the pit would be near-zero, but (assuming a sufficient energy source such as solar or nuclear) could Homo Sapiens thrive at /some/ depth in the pit indefinitely? Or, is the Moon's gravity too weak to bind that atmosphere?

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What would be the atmospheric pressure gradient in an open pit reaching to the centre of the moon?

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