Toric selfdual Einstein metrics on compact orbifolds

We prove that any compact self-dual Einstein $4$-orbifold of positive scalar curvature whose isometry group contains a $2$-torus is, up to an orbifold covering, a quaternion Kähler quotient of $(k-1)$-dimensional quaternionic projective space by a $(k-2)$-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the $4$-orbifold