  A family of quantum fields is said to be strongly local if it generates a
local net of von Neumann algebras. There are few methods of showing directly
strong locality of a quantum field. Among them, linear energy bounds are the
most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot
admit linear energy bounds. In this paper we give a new direct method to prove
strong locality in two-dimensional conformal field theory. We prove that if a
chiral conformal field satisfies an energy bound of degree $d-1$, then it also
satisfies a certain local version of the energy bound, and this in turn implies
strong locality. A central role in our proof is played by diffeomorphism
symmetry. As a concrete application, we show that the vertex operator algebra
given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is
strongly local. For central charge $c > 2$, this yields a new conformal net. We
further prove that these nets do not satisfy strong additivity, and hence are
not completely rational.
