Equidistribution conjecture for dense lattice orbits on compact homogeneous spaces

Let $G$) be a connected semisimple Lie group with finite center, let $\Gamma$ be a lattice in $G$, and let $Y$ be a compact homogeneous space of $G$. Let $K$ be a maximal compact subgroup of $G$, let $d$ be an invariant Riemannian metric on the associated symmetric space, and write $G_T=\{g\in G:d(K,Kg)<T\}$. Equidistribution conjecture. Every dense orbit of $\Gamma$ in $Y$ is equidistributed: there exists a smooth measure $\nu$ on $Y$ such that, for every $y\in Y$ with $\overline{\Gamma y}=Y$ and every Borel set $\Omega\subset Y$ whose boundary has measure zero,

$$\#\{\gamma\in \Gamma:\,\gamma y\in \Omega,\,d(K,K\gamma)<T\}\sim_{T\to\infty}\nu(\Omega)\cdot\frac{\operatorname{Vol}(G_T)}{\operatorname{Vol}(G/\Gamma)}.$$

This would extend the preceding equidistribution theorem from projective homogeneous varieties to arbitrary compact homogeneous spaces; the source provides no resolution of the conjecture.