An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces

It is shown that the growth rate $(\lim_r |B(r)|^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space $\mathbb{H}^d, d>2,$ is at most $k-1-\epsilon$, for an $\epsilon > 0$, depending only on $k$ and $d$. We don't know if there is a universal $\epsilon_u > 0$, such that $k-1-\epsilon_u$ upperbounds the growth rate for any $k$-regular tiling, when $d > 2$?