Contractible-intersection fundamental-domain conjecture for aspherical closed manifolds

Let $X$ be an aspherical closed manifold, let $\widetilde X$ be its universal cover, and let $\pi_1(X)$ act on $\widetilde X$ by deck transformations. A closed fundamental domain is a closed subset $D\subseteq\widetilde X$ whose translates under this action cover $\widetilde X$ in the usual fundamental-domain sense. Contractible-intersection conjecture. The action of $\pi_1(X)$ on $\widetilde X$ admits a closed fundamental domain $D$ such that every nontrivial intersection of translates of $D$ under the $\pi_1(X)$-action is contractible. This would give fundamental domains whose unions have homotopy properties controlled by their intersection pattern; the paper offers it as a reasonable suggestion and gives no resolution.