Compact nonpositively or negatively curved orbifolds that are not very good

Determine whether there exists a compact Riemannian orbifold of nonpositive curvature, or a compact Riemannian orbifold of negative curvature, that is not very good (i.e., does not admit a finite covering by a manifold).

Background

The paper reviews that in nonpositive curvature any complete Riemannian orbifold is good, and proves that complete locally symmetric orbifolds of nonpositive curvature are very good. It also notes Kapovich’s construction of complete negatively curved orbifolds that are not very good, giving a negative answer to a question of Margulis, but these examples are noncompact.

Against this backdrop, the authors ask whether such non-very-good examples can exist in the compact setting for nonpositive or negative curvature, a case not covered by the cited positive results and known constructions.