I have a feeling that this is true (only for positive values in our set, obviously) but I can't prove it or even completely convince myself that it's true.
If not a formal proof (although those are welcome too), some intuition, even geometric/visual, would be great.
Proof, with some parts sketched out with formal derivation left to the reader.
The mean of a set is always less than or equal to the maximum of the set, with equality when all members of the set are equal (proof: the sum of the set is less than or equal to the maximum times the number of elements in the set, divide both sides by the number of elements).
The maximum absolute deviation in a set of positive numbers is less than or equal to the maximum of the set (proof: the mean lies between the maximum and minimum, therefore any absolute deviation from the mean is less than or equal to the difference between the maximum and minimum, which is less than or equal to the minimum).
Therefore, the mean of the set of all absolute deviations is less than the maximum absolute deviation, which is less than the maximum of the set.