  We prove that each of the model structures for ($n$-trivial, saturated)
comical sets on the category of marked cubical sets having only faces and
degeneracies (without connections) is Quillen equivalent to the corresponding
model structure for ($n$-trivial, saturated) complicial sets on the category of
marked simplicial sets, as well as to the corresponding comical model
structures on cubical sets with connections. As a consequence, we show that the
cubical Joyal model structure on cubical sets without connections is equivalent
to its analogues on cubical sets with connections and to the Joyal model
structure on simplicial sets. We also show that any comical set without
connections may be equipped with connections via lifting, and that this can be
done compatibly on the domain and codomain of any fibration or cofibration of
comical sets.
