Contractible-translate fundamental-domain conjecture for aspherical manifolds

Let $M$ be an aspherical manifold, let $\widetilde M$ be its universal cover, and let $\pi_1(M)$ act on $\widetilde M$ by deck transformations. Contractible-translate conjecture. There is a compact set $K\subseteq\widetilde M$ such that the translates of $K$ under the deck action cover $\widetilde M$, and every nonempty intersection of these translates is contractible. This would provide tessellations with contractible intersections beyond the constant-curvature setting; the paper presents it as an interesting possible generalization, with no resolution given.