Can we prove uniqueness under a weaker hypothesis? For instance, if a complete Calabi-Yau metric on $\mathbb{C}^{3}$ is uniformly equivalent to $g_{\mathbb{C}^{3}}$, then does it need to be a member of our family of Taub-NUT type metrics? If we are only given the topology of $\mathbb{C}^{3}$, then is it possible to characterise our Taub-NUT type metrics in terms of its tangent cone at infinity and some extra curvature decay conditions?