The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE_\kappa (both with simple and non-simple loops, i.e., for the whole range \kappa \in (8/3, 8) how to derive the connection probabilities in conformal rectangles for a conditioned version of CLE_\kappa which can be interpreted as a CLE_\kappa with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a conformal square, we prove that the probability that the two wired sides hook up so that they create one single loop is equal to 1/(1 - 2 \cos (4 \pi / \kappa )).
Comparing this with the corresponding connection probabilities for discrete O(N) models for instance indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is CLE_\kappa where \kappa is the value in (8/3,4] such that -2 \cos (4 \pi / \kappa ) is equal to N (resp. the value in [4,8) such that -2 \cos (4 \pi / \kappa ) is equal to \sqrt {q}.
Our arguments and computations build on the one hand on Dub\'edat's SLE commutation relations (as developed and used by Dub\'edat, Zhan or Bauer-Bernard-Kyt\"ol\"a) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field, as recently derived in works with Sheffield and with Qian.
