Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic $3$-space

In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. 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. Therefore, in this paper we examine the doubly truncated Coxeter orthoscheme tilings and the corresponding congruent and non-congruent hyperball packings. We proved that related to the mentioned Coxeter tilings the density of the densest congruent hyperball packing is $\approx 0.81335$ that is -- by our conjecture -- the upper bound density of the relating non-congruent hyperball packings too.