Nonuniqueness of Calabi-Yau metrics with maximal volume growth

We construct a family of inequivalent Calabi-Yau metrics on ${\mathbf{C}}^3$ asymptotic to $\mathbf{C}\times A_2$ at infinity, in the sense that any two of these metrics cannot be related by a scaling and a biholomorphism. This provides the first example of families of Calabi-Yau metrics asymptotic to a fixed tangent cone at infinity, while keeping the underlying complex structure fixed. We propose a refinement of a conjecture of Székelyhidi addressing the classification of such metrics.