  In our paper "Profinite $\infty$-operads", we developed a model for the
homotopy theory of profinite $\infty$-operads in the form of dendroidal objects
with values in profinite sets. In this sequel, we describe a Quillen equivalent
model based on dendroidal objects with values in Quick's model category of
profinite spaces. This model enables us to prove a Dwyer-Kan style
characterization of the weak equivalences between profinite $\infty$-operads
and to characterize the underlying $\infty$-category as an $\infty$-categorical
pro-category. Moreover, we use this model to show that the profinite completion
used by Boavida de Brito-Horel-Robertson in their study of the
Grothendieck-Teichm\"uller group fits in our general theory of profinite
completion as a left Quillen functor.
