One-parameter family conjecture for Calabi–Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C}\times A_2$

Let $\omega$ be a Calabi–Yau metric on $\mathbf{C}^3$, and suppose its tangent cone at infinity is $\mathbf{C}\times A_2$. Metrics are considered up to scaling and isometry. One-parameter family conjecture. Up to scaling and isometry there is a one-parameter family of Calabi–Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C}\times A_2$ at infinity. This predicts nonuniqueness of Calabi–Yau metrics on $\mathbf{C}^3$ when the tangent cone is $\mathbf{C}\times A_2$, contrasting with uniqueness phenomena for simpler tangent cones.