Would a sufficiently deep cavern on Mars provide habitable temperature and atmospheric pressure?

Dick Tracy discovered minimally dressed Moon People living comfortably outdoors on the Moon. Their Far Side of the Moon valley provided enough air pressure for habitability.

[![enter image description here](https://i.stack.imgur.com/Ha4r9.jpg)](https://i.stack.imgur.com/Ha4r9.jpg)

The Dead Sea is 402 m below sea level. Air pressure is measurably higher than at sea level (800 vs. 760 mmHg). The higher partial pressure of O2 actually improves exercise tolerance in patients with lung disease. [https://pubmed.ncbi.nlm.nih.gov/8756971/#:~:text=At%20the%20Dead%20Sea%2C%20located,in%20high%20inspired%20oxygen%20levels](https://pubmed.ncbi.nlm.nih.gov/8756971/#:%7E:text=At%20the%20Dead%20Sea%2C%20located,in%20high%20inspired%20oxygen%20levels)

How deep would a Mars cavern need to be to provide habitable ambient pressure like the 4.5 psi (31 kPa, 232.72 mm Hg) pressure used in ISS EVA suites? Since ISRU advocates propose mining operations it would be elegant if the excavation provided habitat as well.

Mineshaft barometric pressure is a produced by the weight (not mass) of the air above. Gravity decreases with depth, reducing the weight of air at that depth but not the weight of the air above. Gravitational acceleration decreases to zero at the center of a sphere. But it does not decrease the weight of air higher in the shaft, so pressure does not decrease to zero at the center of a planet.

Same for temperature: it decreases the density of the air locally, but not the pressure of the air above.

Temperature increases with depth. On earth, geological thermal gradients are typically 25 K/km. This puts a limit on maximum depth of human presence.

However, one view <https://www.lpi.usra.edu/meetings/geomars2001/pdf/7044.pdf> is that Mars is significantly less thermally active than Earth and has ¼ the thermal gradient. This, along with the lower surface temperature, would permit much deeper habitation. The same paper calculates typical geologic depth to liquid water of 5 to 6 km. This implies a local temperature of (very roughly) 0 °C at that depth.

Can the “Dick Tracy” approach theoretically provide habitable temperature and pressure while also protecting from radiation? And would your wrist phone work at that depth?

Related: [Mars trench](https://space.stackexchange.com/questions/32503/how-deep-a-valley-or-trench-would-be-needed-on-mars-to-provide-the-same-atmosphe?rq=1)