Farrell–Zdravkovska geometric conjecture on bounding flat and almost flat manifolds

Let $M^n$ be a closed manifold. In part (a), assume that $M^n$ is a flat Riemannian manifold. In part (b), assume that $M^n$ admits an almost flat structure. Let $W^{n+1}$ be a compact manifold with boundary $M^n$, and write $W\setminus\partial W$ for its interior. Farrell–Zdravkovska geometric conjecture. (a) If $M^n$ is a flat Riemannian manifold, then $M^n=\partial W^{n+1}$ for some $W^{n+1}$ whose interior supports a complete finite-volume hyperbolic structure. (b) If $M^n$ supports an almost flat structure, then $M^n=\partial W^{n+1}$ for some $W^{n+1}$ whose interior supports a complete finite-volume Riemannian metric with negative sectional curvatures. This is a stronger geometric form of the question whether flat or almost flat manifolds bound. The source presents it as a conjecture attributed to Farrell and Zdravkovska and supplies no evidence of resolution.