The degree-two homogeneous perturbation conjecture for polynomial automorphisms

Let $k$ be a field, let $I$ denote the identity map on affine $n$-space, and let $H$ be a homogeneous polynomial map of degree $2$. Thus $F=I+H$ is a polynomial automorphism in $\operatorname{GA}_n(k)$.

Degree-two tameness conjecture. If

$$F=I+H\in \operatorname{GA}_n(k),$$

where $H$ is homogeneous of degree $2$, then $F$ is tame.

The conjecture is motivated by the paper's computational theorem establishing tameness in several low-degree and small-finite-field cases. No general proof or counterexample is given.