The geometric fibration conjecture for reductive homogeneous spaces

Let $G/H$ be a reductive homogeneous space, with maximal compact subgroup $K$ and $K_H=K\cap H$, and set $X=K/K_H$. Let $\Gamma$ be a torsion-free subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. Then $M=\Gamma\backslash G/H$ is the associated compact quotient.

Geometric fibration conjecture. The group $\Gamma$ is the fundamental group of a closed aspherical manifold $M$, and there exists a smooth $\Gamma$-equivariant fiber bundle

$$G/H\longrightarrow \widetilde{M}$$

whose fibers are $G$-translates of $X$, where $\widetilde{M}$ is the universal cover of $M$.

This conjecture predicts that compact quotients virtually fiber with fibers modeled on the maximal compact subspace. It is known in several examples, including certain group manifolds and $\operatorname{SO}(2n,2)/\operatorname{U}(n,1)$, but remains open in general.