Geometric counting conjecture for homogeneous spaces

Let $G$ be a connected Lie group and $H<G$ a closed subgroup such that $Z:=G/H$ carries a $G$-invariant positive Radon measure $\mu_Z$ satisfying the measure disintegration formula with respect to Haar measures $\mu_G$ and $\mu_H$. Assume that $H$ has finitely many connected components. Let $\Gamma<G$ be a lattice with

$$\operatorname{vol}(G/\Gamma)=1.$$

Set $\Gamma_H:=\Gamma\cap H$, and assume that it is a lattice in $H$ with

$$\operatorname{vol}(H/\Gamma_H)=1.$$

Homogeneous-space counting conjecture. There exists an exhausting compact family $\mathcal{B}^Z=(B_R^Z)_{R>0}$ of $Z$ such that the quadruple $(Z,\mu_Z,\Gamma/\Gamma_H,\mathcal{B}^Z)$ satisfies the monotone-counting property (MTC). This asserts the expected asymptotic counting of the discrete set $\Gamma/\Gamma_H$ in the exhausting family $\mathcal{B}^Z$ relative to the measure $\mu_Z$. The claim is presented as an open problem in the source; the existence of such a family is not established there.