Alexander's perimeter conjecture for intersections of congruent disks

Let finitely many congruent disks have centers in the Euclidean plane, and let a contraction mean that every pairwise distance between the new centers is no greater than the corresponding distance between the original centers. Consider the intersection of the disks before and after the contraction.

Alexander's conjecture. Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks cannot decrease.

This conjecture would sharpen results of Bezdek and Connelly on intersections of congruent disks. The source presents it as an open question attributed to Alexander, and gives no resolution.