Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in ${\mathbb{R}}^{n+1}$

We study the asymptotic distribution of norm ball averages along orbits of a lattice $\mathrm{\Gamma }\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of ${\mathbb{R}}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in ${\mathbb{R}}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.