Asymptotic product conjecture for the Calabi ansatz metric on $\mathbb C^k$

Let $\omega$ be the metric on $\mathbb C^k$ determined by Lemma, and let $\omega_p=iD_0\partial\bar{\partial}s^p$, where $s=|z|^2$ on $\mathbb C^k$. Say that a metric is asymptotic to $\omega_p$ when it satisfies the conditions of Definition. Asymptotic product conjecture. The metric $\omega$ is asymptotic to $\omega_p$ in the sense of Definition. The preceding argument establishes this conclusion in the treated case, and the surrounding discussion suggests that the same conclusion should hold for arbitrary values of $k$; the general case remains unproved in the supplied text.