Moduli-space conjecture for the canonical embedding of holomorphic automorphisms

For $n\geq 1$, let $\operatorname{Aut}_{\operatorname{hol}}(\mathbb C^n)$ be the group of holomorphic automorphisms of $\mathbb C^n$, and define

$${\frak{X}}^n(F)=\left(F(0),DF(0),L\left(\log\left\|DF(0)^{-1}(F-F(0))\right\|\right)\right).$$

The map ${\frak{X}}^n$ is injective, and $\operatorname{Ran}({\frak{X}}^n)$ denotes its range. A moduli space is a proto-moduli space whose membership admits an intrinsic description not referring to the original group or embedding.

Moduli-space conjecture. $\operatorname{Ran}({\frak{X}}^n)$ is a moduli space for all $n\geq 1$.

The preceding discussion explicitly identifies the range as a moduli space when $n=1$; the assertion for $n>1$ is the corresponding unresolved extension. This statement restates the preceding conjecture in the paper and is therefore merged with it.