Kähler-Ricci shrinker asymptotic conicity conjecture

A Kähler-Ricci shrinker is a complete shrinking gradient Kähler-Ricci soliton $(X,J,\omega,f)$ satisfying

$$\operatorname{Ric}(\omega)+\sqrt{-1}\partial\bar\partial f=\omega,$$

with $J\nabla f$ a holomorphic Killing field. A Kähler cone metric on $Y$ is the proposed asymptotic model at infinity.

Asymptotic conicity conjecture. A Kähler-Ricci shrinker is weakly asymptotically conical in the sense that it has a unique asymptotic cone at infinity, which is a Kähler cone metric on $Y$.

The conjecture concerns the metric geometry of non-compact Kähler-Ricci shrinkers without prescribing behavior at infinity. It is known under quadratic curvature decay, but remains open in general.