  We prove that an operator system is (min, ess)-nuclear if its C*-envelope is
nuclear. This allows us to deduce that an operator system associated to a
generating set of countable discrete group by Farenick et al. is (min,
ess)-nuclear if and only if the group is amenable. We also make a detailed
comparison between ess and other operator system tensor products and show that
an operator system associated to a minimal generating set of a finitely
generated discrete group (resp., a finite graph) is (min, max)-nuclear if and
only if the group is of order less than or equal to 3 (resp., every component
of the graph is complete).
