On the torsion in the center conjecture

We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conjecture that every almost nonnegatively curved closed \begin{document}$ m $\end{document} -dimensional manifold \begin{document}$ M $\end{document} admits a finite cover \begin{document}$ \tilde M $\end{document} for which the number of leafs is bounded in terms of \begin{document}$ m $\end{document} such that the torsion of the fundamental group \begin{document}$ π_1 \tilde M $\end{document} lies in its center.