  In this paper we consider ball packings in $4$-dimensional hyperbolic space.
We show that it is possible to exceed the conjectured $4$-dimensional
realizable packing density upper bound due to L. Fejes T\'oth (Regular Figures,
1964). We give seven examples of horoball packing configurations that yield
higher densities of $0.71644896\dots$, where horoballs are centered at ideal
vertices of certain Coxeter simplices, and are invariant under the actions of
their respective Coxeter groups.
