  We prove that the Ahlfors regular conformal dimension is upper semicontinuous
with respect to Gromov-Hausdorff convergence when restricted to the class of
uniformly perfect, uniformly quasi-selfsimilar metric spaces. Moreover we show
the continuity of the Ahlfors regular conformal dimension in case of limit sets
of discrete, quasiconvex-cocompact group of isometries of uniformly bounded
codiameter of $\delta$-hyperbolic metric spaces under equivariant pointed
Gromov-Hausdorff convergence of the spaces.
