Do ballistic parachutes work at high altitudes in thin air?

We understand that ballistic parachutes for GA have not been deployed at high altitudes, only the military have done that, apparently successfully. According to BRS, in thin air there would be a longer free fall until the chute opens. BRS says their parachute typically needs a 2000 ft. fall to fully open.

However, the GFC will be flown at ≥ 4000 feet MSL with generally ≥ 3500 feet AGL most of the course. More.

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