A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are very limited 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. 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.