  Through a theoretical coupled wire model, we construct strongly correlated
electronic \emph{integer} quantum Hall states. As a distinguishing feature,
these states support electric and thermal Hall transport violating the
Wiedemann-Franz law as
$\left(\kappa_{xy}/\sigma_{xy}\right)/\left[\left(\pi^{2}k_{B}^{2}T\right)/3e^{2}\right]<1$.We
propose a new Abelian incompressible fluid at filling $\nu=16$ that supports a
bosonic chiral $(E_{8})_{1}$ conformal field theory at the edge and is
intimately related to topological paramagnets in (3+1)D. We further show that
this topological phase can be partitioned into two non-Abelian quantum Hall
states at filling $\nu=8$, each carrying bosonic chiral $(G_{2})_{1}$ or
$(F_{4})_{1}$ edge theories, and hosting Fibonacci anyonic excitations in the
bulk. Finally, we discover a new notion of particle-hole conjugation based on
the $E_{8}$ state that relates the $G_{2}$ and $F_{4}$ Fibonacci states.
