The ambiguities arise in the definition of spin networks(coloring, vertex, ...). It's by no means clear that there is one unique spin network. What the paper says is that the ambiguities in the connection show up as ambiguities in the definition of the spin network, i.e. that there is a class of spin networks instead of simply one.
In addition it is not clear how to define the measure. The SF is something like a Lagrangian PI and is therefore a derived object. The fundamental object is the canonical PI - which is not known due to the ambiguities in the Hamiltonian. Therefore the measure is not known.
What Alexandrov says is that during the construction of the PI there are several choices how to proceed. Looking at one specific result there is no obviously visible ambiguity. But this is due to restrictions and choices during the construction.