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#05AB1E (legacy), (削除) 26 (削除ここまで) 24 bytes

Δ©ÅA®PYzmY®zO/®nÅAt)4V}н

Try it online or see the steps of all test cases.

Explanation:

Δ # Loop until the list no longer changes:
 © # Store the current list in the register (without popping)
 # (which is the implicit input-list in the first iteration)
 ÅA # Calculate the arithmetic mean of this list
 ®P # Take the product of the list
 Yz # And 1/`Y`
 m # Then take the result of the product to the power of 1/`Y`
 # (NOTE: a^(1/4) is the same as 4√a)
 ®z # Get 1/x for each value x in the list
 O # Take the sum of that
 Y / # And then divide `Y` by that sum
 ®n # Take 2 to the power of each number x in the list
 ÅA # Calculate the arithmetic mean of this list
 t # And then take the square-root of that
 ) # Wrap all four results into a list
 4V # And set variable `Y` now to 4 instead of 2
}н # After the list no longer changes: pop and push its first value
 # (which is output implicitly as result)