There is a well known problem of LeBrun-Salamon: are there any non-symmetric compact quaternionic-Kahler manifolds of positive scalar (and Ricci) curvature? It is hard and still unsolved: Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?
The symmetric compact quaternionic-Kahler manifolds ("Wolf spaces") are understood and classified. However, for each Wolf space there is the dual symmetric space, say, $G/H$, which is quaternionic-Kahler of negative scalar (and Ricci) curvature.
For any lattice $\Gamma\subset G$, the double quotient $\Gamma\backslash G/H$ is a locally symmetric quaternionionic-Kahler orbifold of negative curvature and finite volume. However, it can have cusp points, and then it is non-compact.
Are there any compact locally symmetric quaternionionic-Kahler orbifolds? Manifolds? I could not find a reference to either existence or non-existence results.