  We prove that for a quasiself-similar and arcwise connected compact metric
space all three known versions of the conformal dimension coincide: the
conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular
conformal dimension. This answers a question posed by Mathav Murugan.
Quasisimilar spaces include all approximately self-similar spaces. As an
example, the standard Sierpi\'nski carpet is quasiself-similar and thus the
three notions of conformal dimension coincide for it.
  We also give the equality of the three dimensions for combinatorially
$p$-Loewner (CLP) spaces. Both proofs involve using a new notion of
combinatorial modulus, which lies between two notions of modulus that have
appeared in the literature. The first of these is the modulus studied by Pansu
and Tyson, which uses a Carath\'eodory construction. The second is the one used
by Keith and Laakso (and later modified and used by Bourdon, Kleiner,
Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches,
we gain the flexibility of giving upper bounds for the new modulus from the
Pansu-Tyson approach, and the ability of getting lower bounds using the
Keith-Laakso approach. Additionally the new modulus can be iterated in
self-similar spaces, which is a crucial, and novel, step in our argument.
