  We formulate a classification conjecture for conformally invariant families
of measures on simple loops that builds on a conjecture of Kontsevich and
Suhov. The main example in this class of objects was constructed by Werner as
boundaries of Brownian loops. We present partial results towards the algebraic
step of this classification.
  Solving this conjecture would provide another argument explaining why planar
statistical mechanics models with conformally invariant scaling limits
naturally occur in a one-parameter family, together with the dynamical
characterization of SLE via Schramm's central limit argument, and with the
conformal field theory point of view and its central charge parameter.
