This problem dates from the mid-1950's. The planar case was solved by Bezdek and Connelly in 2003, who also showed that the area of the intersection does not increase, and that the result holds even if the disks have unequal radii. In higher dimensions the problem remains open.
The conjecture is known to hold if the rearrangement can be executed by a continuous motion such that the distance between every pair of centers monotonically increases throughout the motion.
*[BC] K. Bezdek and R. Connelly, Pushing disks apart: The Kneser-Poulsen conjecture in the plane, J. Reine Angew. Math. 553 (2002), 221--236.
*[K] M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955), 382--390.
*[P] E. T. Poulsen, Problem 10, Math. Scand. 2 (1954), 346.
* indicates original appearance(s) of problem.