  In \cite{Sz11} we have generalized the notion of the simplicial density
function for horoballs in the extended hyperbolic space $\bar{\mathbf{H}}^n,
~(n \ge 2)$, where we have allowed {\it congruent horoballs in different types}
centered at the various vertices of a totally asymptotic tetrahedron. By this
new aspect, in this paper we study the locally densest horoball packing
arrangements and their densities with respect to totally asymptotic regular
tetrahedra in hyperbolic $n$-space $\bar{\mathbf{H}}^n$ extended with its
absolute figure, where the ideal centers of horoballs give rise to vertices of
a totally asymptotic regular tetrahedron.
  We will prove that, in this sense, {\it the well known B\"or\"oczky density
upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^n$ does not
remain valid for $n\ge4$,} but these locally optimal ball arrangements do not
have extensions to the whole $n$-dimensional hyperbolic space. Moreover, we
determine an explicit formula for the density of the above locally optimal
horoball packings, allowing horoballs in different types.
