4 problems

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

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

Compact quotient–local fibration conjecture. The following are equivalent:

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