  Building off of techniques that were recently developed by M. Carrasco, S.
Keith, and B. Kleiner to study the conformal dimension of boundaries of
hyperbolic groups, we prove that uniformly perfect boundaries of John domains
in the Riemann sphere have conformal dimension equal to 0 or 1. Our proof uses
a discretized version of Carrasco's "uniformly well-spread cut point"
condition, which we call the discrete UWS property, that is well-suited to deal
with metric spaces that are not linearly connected. More specifically, we prove
that boundaries of John domains have the discrete UWS property and that any
compact, doubling, uniformly perfect metric space with the discrete UWS
property has conformal dimension equal to 0 or 1. In addition, we establish
other geometric properties of metric spaces with the discrete UWS property,
including connectivity properties of their weak tangents.
