In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more properties of such a map). They also say that such a manifold $N$ has "non-positive" curvature.


My question is whether one can chose $N$ to be negatively curved, or at least that $N$ has word-hyperbolic fundamental group.
Or, if the requirement that $N$ is aspherical is too strong, whether every manifold is dominated by a manifold with word-hyperbolic fundamental group.



Hyperbolization with word-hyperbolic fundamental group
Charney-Davis in [Strict hyperbolization](https://www.sciencedirect.com/science/article/pii/004093839400027I) showed how to make $N$ locally CAT($-1$), provided $M$ is PL.


Ontaneda in [Riemannian hyperbolization](https://link.springer.com/article/10.1007/s10240-020-00113-1) showed how to make $N$ a Riemannian manifold of negative sectional curvature, provided $M$ is smooth.


Incidentally, just like in the paper of Davis-Januszkiewicz, $N$ and $M$ are closed manifolds.


