Is the packing of the plane by disks of radius 1/2 centered at the points of ${\bf Z} \times {\bf Z}$ "locally rigid" in the sense that no finite subcollection of the disks admits any joint infinitesimal deformations subject to the locations of the other disks?

Certainly it is clear that if you fix the locations of all but 1 of the disks, the 1 "free" disk cannot move in any direction. But it is less clear to me that for all $n \geq 1$, if you fix the locations of all but $n$ of the disks, there is no way for those $n$ disks to move in some joint fashion. Indeed, we do know that when $n$ is large enough, and the $n$ free disks form a sub-cluster of the packing of a suitable kind, there are lots of ways to rearrange those $n$ disks, if we either remove the disks from the plane and then replace them, or if we allow the disks to temporarily intersect the fixed disks on the way to their respective destinations.