The generation conjecture for three-dimensional polynomial automorphisms fixing one variable

Let $\mathbb{K}$ be the field appearing in the polynomial automorphism group $\operatorname{GA}_3(\mathbb{K})$. An automorphism fixing one variable is a polynomial automorphism that leaves one coordinate variable unchanged. Generation conjecture. The group

$$\operatorname{GA}_3(\mathbb{K})$$

is generated by the automorphisms fixing one variable. This is presented as one of several conjectures about generating sets of polynomial automorphism groups; over fields such as $\mathbb{F}_4$ and $\mathbb{F}_8$, the surrounding discussion notes that every automorphism fixing one variable is even, so an odd automorphism would disprove the conjecture.