Folklore rigidity conjecture for AC Ricci-flat Kähler metrics on affine space

Let $g$ be a complete asymptotically conical Ricci-flat Kähler metric on $\mathbb{C}^n$, asymptotic to a Kähler cone $\mathcal{C}$ whose link is smooth. The AC metric rigidity conjecture. Then $g$ is the flat Euclidean metric. This conjecture predicts rigidity of complete AC Ricci-flat Kähler metrics on $\mathbb{C}^n$ when the asymptotic cone has a smooth link; the paper proves the assertion for $n=3$, while the general case remains open.