Székelyhidi's parametrization conjecture for Calabi–Yau metrics with tangent cone $\mathbf{C} \times A_2$

Let $\mathbf{C}^3$ carry a Calabi–Yau metric whose tangent cone at infinity is $\mathbf{C} \times A_2$. Identify metrics that differ by biholomorphism and scaling. Székelyhidi's parametrization conjecture. The resulting space of metrics is parametrized by

$$\mathbf{C}/S^1 \cong \mathbf{R}_{\ge 0}.$$

This refines a conjecture of Székelyhidi concerning the moduli of Calabi–Yau metrics with prescribed tangent cone at infinity. The source does not state a resolution, so the conjecture remains open.