The compact quotient implies local geometric fibration conjecture

Let $G/H$ be a reductive homogeneous space. A local geometric fibration is a smooth fiber bundle whose total space is an open subset of $G/H$ and whose fibers are translates of the maximal compact subspace $X=K/K_H$.

Compact quotient implies local fibration conjecture. If $G/H$ admits compact quotients, then it admits local geometric fibrations.

This is the implication from compact quotients to local geometric fibrations in the broader equivalence conjecture. The paper presents it as a consequence predicted by the geometric fibration conjecture and as a necessary condition in its results.