Distribution of facets of higher-dimensional Klein polyhedra

We consider facets of Klein polyhedra of a given integer-linear type $\mathscr T$ in a certain lattice. Let $E_\mathscr T(N,s)$ be the typical number of facets, averaged over all integral $s$-dimensional lattices with determinant $N$. Assume that the interior of any facet of type $\mathscr T$ contains at least one point of the corresponding lattice. We prove that
$$E_\mathscr T(N,s)=C_\mathscr T \ln^{s-1}N+O_\mathscr T (\ln^{s-2} N \cdot \ln\ln N) \quad\text{as } N \to \infty,$$
where $C_\mathscr T$ is a positive constant depending only on $\mathscr T$.
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