  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 Sz\'ekelyhidi addressing the
classification of such metrics.
