Tangent-cone conjecture for moduli space

Let $\M_g$ be moduli space with the Teichmüller metric, and let ${\cal C}_g$ be the curve complex of a closed genus-$g$ surface: its vertices are nontrivial, nonperipheral isotopy classes of simple closed curves, with simplices given by collections admitting mutually disjoint representatives. Define the tangent cone at infinity by

$$\operatorname{cone}(\M_g):=\lim_{n\to\infty}(\M_g,\tfrac1n d_{\rm Teich}),$$

with pointed Gromov–Hausdorff convergence. Tangent-cone conjecture for moduli space.

$$\operatorname{cone}(\M_g)\text{ is homeomorphic to the open cone on }{\cal C}_g/{\rm Mod}_g.$$

The source notes that the precise formulation requires orbi-complexes and that related arithmetic-lattice analogues are known; the stated moduli-space identification remains conjectural.