Global bubbling conjecture for Kähler–Einstein orbifold smoothings

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a smoothing of a Kähler–Einstein orbifold $(X_0,\omega_0)$ with $A_k$-singularities, and suppose that $\operatorname{Aut}(X_0)$ is discrete. The nearby fibers $X_t$ are the smoothings under consideration, and their local deformation data determine the associated bubble trees. Global bubbling conjecture. Nearby $X_t$ admits Kähler–Einstein metrics whose full multiscale bubble tree can be recovered just from local algebraic data of the given family. More precisely, by considering the curve induced by the family in the versal deformation space of the $A_k$-singularity, and by varying sections of the family passing through the singularity, the tree of pointed Gromov–Hausdorff limits at a given singularity matches the local description given in the paper's theorem on $A_k$ limits. This is proposed as a global construction extending the local bubbling analysis for $A_k$-singularities; the expected proof is based on gluing techniques for constant-scalar-curvature Kähler metrics and multiscale analysis of non-collapsed Einstein $4$-manifolds.