Disk patterns, quasi-duality and the uniform bounded diameter conjecture

We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a connection between the diameter of the skinning image and certain discrete extremal width on the Coxeter graph of the reflection group.