The desingularization conjecture for singular spherical and hyperbolic orbifolds

Let $(M_o,\mathbf{g}_o)$ be a singular spherical or hyperbolic compact orbifold. A sequence of smooth Einstein metrics is said to converge to $(M_o,\mathbf{g}_o)$ in the Gromov–Hausdorff sense when the underlying metric spaces converge in Gromov–Hausdorff distance. Desingularization conjecture. The orbifold $(M_o,\mathbf{g}_o)$ is not a limit of smooth Einstein metrics in the Gromov–Hausdorff sense. The paper proves this claim for spherical or hyperbolic $4$-orbifolds having a singularity of type $\mathbb{R}^4/\mathbb{Z}_2$, while the more general formulation stated here is presented as an expected extension and requires dealing with trees of singularities.