  In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing
exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into
truncated tetrahedra. Therefore, in order to get a density upper bound for
hyperball packings, it is sufficient to determine the density upper bound of
hyperball packings in truncated simplices. In this paper we prove, using the
above results and the results of papers \cite{M94} and \cite{Sz14}, that the
density upper bound of the saturated congruent hyperball (hypersphere) packings
related to the corresponding truncated tetrahedron cells is realized in a
regular truncated tetrahedra with density $\approx 0.86338$. Furthermore, we
prove that the density of locally optimal congruent hyperball arrangement in
regular truncated tetrahedron is not a monotonically increasing function of the
height (radius) of corresponding optimal hyperball, contrary to the ball
(sphere) and horoball (horosphere) packings.
