  In this paper we study $\times_0$-products of Lann\'er diagrams. We prove
that every $\times_0$-product of at least four Lann\'er diagrams with at least
one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that
known classifications exhaust all compact hyperbolic Coxeter polytopes that are
combinatorially equivalent to products of simplices.
  We also consider compact hyperbolic Coxeter polytopes whose every Lann\'er
subdiagram has order $2$. The second result of this paper slightly improves
recent Burcroff's upper bound on the dimension of such polytopes to $12$.
