Hyperbolization with word-hyperbolic fundamental group

Question

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.

Answer 1

Charney-Davis in Strict hyperbolization showed how to make $N$ locally CAT($-1$), provided $M$ is PL.

Ontaneda in Riemannian hyperbolization 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.