The precise geometric fibration conjecture for compact quotients

Let $G/H$ be a reductive homogeneous space, with maximal compact subgroup $K$. Let $\pi_H:G/K_H\to G/H$ and $\pi_K:G/K_H\to G/K$ be the natural projections, and let $\dim_+(G/H)$ denote the dimension specified in the source. Let $\Gamma$ be a torsion-free subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$.

Precise geometric fibration conjecture. There exists a $\Gamma$-invariant contractible smooth submanifold $\widetilde{M}$ of $G/K$, of dimension $\dim_+(G/H)$, such that

$$G/H=\bigsqcup_{\widetilde{m}\in\widetilde{M}}\pi_H\circ\pi_K^{-1}(\widetilde{m}).$$

This is presented as a more precise form of the geometric fibration conjecture. The relation between compact quotients and the proposed fibration structure remains open in general.